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STEREOTOMY 





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NrOTES ON STEREOTOMY, 

PREPARED FOR THE USE OF STUDENTS 

IN 

CIVIL ENSINEERING 

IN THE 

MASSACHUSETTS INSTITUTE OF TECHITOLOGY, 

DWIGHt' PORTER. 
PLANE, SINGLE-CURVED AND DOUBLE-CURVED SURFACES. 



Copyright, 1895, by Dwight Porter. 



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STEREOTOMY. 

1. Introductory . Stereotomy is defined as "the science or art of 
cutting solids into certain figures or sections. " It thus includes, and 
the term is sometimes used as synonymous with, the art of cutting stones 
for masonry structures. This application only will here be considered. 
Pull knowledge of all the minor details of practical stone-cutting is not 
essential for the engineer. He should, however, be competent to prepare 
whatever drawings and patterns may be required in any masonry construc- 
tion, and should understand the mode of procedure to be followed in 
bringing the stones to shape. The proportions of the structure are here 
assumed to have been fixed by the designer, with reference to the laws 

of stability and of architecture, and the study to be iindertaken deals 
simply with the sub-division into stones, and the preparation of the 
necessary general and detail drawings. Most of the problems here pre- 
sented, with their accompanying plates,^ have been taiken from Trait e' 
Pratique de la Ooupe des Pierres , par Emile Lejexme . 

2. Drawin gs . The nxomber and character of drawings required for a 
stone structure evidently must be governed by circumstances. Ordinarily 
there must be a plan, an elevation and a section, at least. If the 
structure is complicated the drawings become correspondingly numerous, 
and comprise elevations from various points of view, together with sundry 
plans, sections and other details. If definite sizes of stones are to 

be required, and a particular arrangement in the structure, covirse plans 
showing such arrangement in detail must be prepared. 

General drawings should commonly be made, if practicable, to as 
large a scale as one-half or three-quarters of an inch to the foot; 
detail drawings to as large a scale as is necessary for clearness, and 
frequently they must be of full size. 

All dimensions for stone-cutters and masons are to be expressed in 
feet, inches, halves, quarters, etc., to the nearest l/4 or 1/8 of an 
inch, and in some cases to the nearest l/l6 or l/32 of an inch. 

If a stone is to be rectangular in shape, a simple sketch with 
dimensions may be sufficient guide to the stone-cutter; but if irregular, 
sundry sketches, an isometric view, etc., may be needed. Patterns 
(moulds) and templets made of wood, zinc or other material, to corre- 
spond to various faces, sections or profiles, may also be required. To 
prepare these, full-size drawings of the structure are made on a large 
floor or other surface. For structures of moderate size the instruments 
needed for such drawings are mainly large beam compasses, a long wooden 
straight-edge and a large steel square; but for the largest drawings it 
may be necessary to use cords or wires in determining points or lines. 
If the stone is vmusually complicated in shape or involves carving, a 
full-size or half-size model may be necessary. For such work as an 
emblematic Resign, the original designer first prepares a sketch; the 
sculptor then constructs a clay model, using the sketch as a basis, and 
generally modifying it somewhat according to his own taste; from the 
clay model a plaster cast is made, and from this the stone-cutter works, 
transferring points from cast to stone as frequently and with as great 
accuracy as the case demands. 

-1- 



Drawings for masonry are commonly made without reference to the 
thickness of joints, and where necessary in order to prevent misunder- 
standing, especially in the case of detail drawings, a note should be 
added to the effect that "in this drawing no allowance has been made for 
Joints. " The thickness assigned on the drawing to any course is inter- 
preted by the stone-cutter as the distance from the top of one stone to 
the top of the next lower one, as laid. (See Pig. 1) 

The drawings must give all necessary dimensions, and must also make 
it entirely clear which surfaces are to be exterior and visible when the 
stone is in place and which are to be Joint surfaces, the style of finish 
to be given the faces, and the thickness to be assumed for Joints. The 
date of the drawing, and in some cases the name or initials of the 
draftsman, should also be added. In dimension work a system of numbering 
courses and stones, for convenience in subsequent identification, is 
generally adopted. The accompanying plates, reduced from the original 
drawings, will serve to Illustrate some of the features to which refer- 
ence has been made. 

In any drawing discretion must be exercised as to what lines or 
portions of the structure it is best to show in that particular view, 
and what to omit, and the draftsman should not allow himself to be 
hedged in by unvarying rules in this matter. The object should be to 
make it clear and unmistakable what general form or what details, as the 
case may be, are proposed for the sttnacture in question. Lines which in 
any particular view are necessary to this end should be given; those 
which are not, or which serve to confuse the drawing, or which can be 
better shown in another view, may be omitted. The following principles 
are commended to the notice of the student (See "Practical Hints for 
Draftsmen, " MacCord) :- 

(a) "In each separate view whatever is shown at all should be rep- 

resented in the most explanatory manner. " 

(b) "That which is not explanatory, in any one view, may be omitted 

therefrom, if sufficiently defined in other views. " 

(c) "The proper position of a cutting plane is that by which the 

most information can be clearly given." 

(d) "It is not necessary to show in section everything which might 

be divided by the cutting plane." 

(e) "TOiatever lies beyond a cutting plsuie may be omitted when no 

necessary information would be conveyed by its representation. " 
A section or development may be made on a broken, or even upon a 
curved line, if anything is to be gained by so doing. For a figure 
symmetrical about a centre line, it is often sufficient, except for a 
picture drawing, to show only one half in any one view, a half plan or 
half elevation sometimes being combined with a half section. 

So far as making the drawings is alone concerned, Stereotomy is 
for the student an application of Descriptive Geometry, the problems 
most frequently arising relating to the intersection and development of 
surfaces. Ihen accurate execution is required the student should be very 
careful in laying out the main lines of his drawing, particularly as 
regards dimensions, parallels and perpendiculars. Economy of labor and 
quickness of work must constantly be aimed at and, above all, the great- 
est care must be taken in accurately and clearly noting dimensions on 
the drawing. The directions given below will aid in obtaining satisfac- 
tory results:- 

-2- 



(a) In laying off by scale a number of successive distances along 
a line, lay off the successive sums of the distances, referred 
in each case to the same starting point, rather than lay off 
one distance after another, referring each time to the point 
last fixed. 

(b) In laying off a distauice too short to be easily given by the 
dividers, set back from the reference point any arbitrary dis- 
tance, and then set forward from the point thus fixed the 
assumed distance plus the one required. 

(c) To draw a parallel to a given line, when the T square or tri- 
angles are not applicable, strike arcs at the extremities of 
the given line and make the required line tangent to the arcs. 

(d) Lines which are to determine a point by their intersection 
should preferably not meet at, a very acute angle, and points 
which are to fix a line should preferably be chosen at a dis- 
tance apart greater than the required length of line. 

(e) The drawing should be carefully verified by the draftsman, 
and advantage taken of all convenient methods of testing its 
accuracy. 

(f ) The pencilling of the drawing should be made complete and 
accurate before inking is begion. 

3. Tracings on Cloth . Tlie usual puirpose of a tracing on cloth is 
to permit the duplication of drawings by the blue process. Pine lines 
should therefore be avoided and for the colors of lines only black or 
vermilion should be used. The student will find the rough side of the 
cloth the more satisfactory to use, and if, on account of slight greasi- 
ness, it does not take ink well, the trouble may be remedied by first 
dusting the surface frith powdered chalk, which is to.be brushed off 
before inking. Erasures may be made with a sharp knife blade, taking 
care not to gouge into the cloth. 

4. Reduction of Drawings by Photography. If drawings are to be 
reduced by photography care must be taken that the lines, letters and 
figures in the original are coarse enough to endure the proposed reduc- 
tion without becoming indistinct. With this precaution observed, the 
general appearance of a drawing will be improved by reduction, owing 

to the greater sharpness of the lines and the disappearance of minor 
imperfections. If dimensions are to be scaled from the reduced copy 
a scale evidently must be constructed upon the original sheet which shall 
be reduced in the same proportion as the drawing itself. As regards the 
lettering of drawings for photo- reproduction see an article by Charles 
W. Reinhardt in Engineering News of June 13, 1895. 

To ivoid the inconvenience of handling large blue prints in the 
field or shop, copies of the original drawing or tracing, reduced by 
photography, are sometimes used. These may to advsintage be motmted 
upon cardboard and varnished. So-called "black prints" can be success- 
fully copied and reduced by photography. A method of copying, to the 
number of forty or more duplicates, drawings made in colored lines or 
washes, the colors being reproduced in the copies, is described in 
Engineering News, Aug. 24, 1893, p. 160. 

-3- 



5. Proportions and Shape of Stones . The depths to be given the 
various courses and the proportions to be observed in the dimensions of 
the stones, sire determined with regard to strength, appearance and 
economy. Tolerably large stones, properly arranged, bind the whole work 
well together and tend to secure uniform distribution of pressure. No 
commensurate advantage is ordinarily to be gained, however, from excess- 
ive size, while on the other hand greatly increased difficulty and cost 
in handling are involved, and there is also danger of the cracking of 
very long stones, owing to the stresses from iwiequal settlement in the 
mass of masonry. By requiring courses to be of uniform depth and the 
stones to be of unifonn size, an appearance of great regularity may be 
gained for the face of the structure; but to insist upon such uniformity 
imposes upon the quarryman limitations which materially Increase the cost 
of the work. Economy will therefore result from allowing, when possible, 
a moderate rsmge in thickness of courses, as well as in other dimensions 
of the stones. A variation in depth of courses, these being arranged to 
diminish regularly in thickness from bottom to top of wall, is also in 
accordance with good taste. 

In engineering structures the conanon range in thickness of courses 
is from 10 or 12 up to 30 inches. Stones are usually required to be at 
least one and one-fourth or even one and one-half times as wide as deep. 
They are allowed a length as great as from three to six times the depth, 
varying accordingly as the material is weak or strong. In copings, 
parapets and other special features of stnictures, where the stones are 
not exposed to heavy pressure from above, it is not essential, however, 
to adhere to the proportions which have been indicated. 

The bed joints of a stone should be made normal, or nearly so, to 
the pressure to be sustained; the other joints should be arranged so 
far as possible to produce right angles and to give the stone a simple 
form. Acute angles are objectionable because they are likely to lead to 
the cracking of the stone. Two distinct bearing surfaces for the same 
stone should also be avoided, since in case of poor fitting and conse- 
quent unequal bearing there is danger of cracking. 

6. Thickness of Joints . The thickness of joints should be con- 
sistent with the general standard aimed at in the masonry, being properly 
least for work which is to be of the highest class as regards strength 
and finish. The requirement of thin joints involves a nicer dressing of 
the joint surfaces of the stones than would otheirwise be given. 

In engineering structures presenting the best type of work, such as 
arches, waterways, piers, etc., the joints are made from one-fourth inch 
to one-half inch thick. In heavy work of a less high grade the joints 
range from one-half inch to one inch in thickness. In fine work in 
building construction joints as thin as one-eighth inch or even less are 
employed. In brick work of a good class the mortar joints are from one- 
eighth inch to three-eighths inch in thickness, and in coarser work 
one-half inch is considered allowable. 

Since the bed-joint surfaces of the stones receive and transmit the 
downward pressures, they should be dressed substantially tme throughout. 
There is not the same necessity for this, however, with the vertical 
joint surfaces, and frequently specifications allow a moderate increase 
in thickness of vertical joints at a certain distance, say 12 inches, 
back from the face of the wall, and even make a distinction as to thick- 

-4- 



nes8 between the two sets of joints at the face of the structure. In 
speaking of the thickness of a joint, either the space or the mortar 
filling between two adjacent stones is evidently in mind; but in practice 
the tenn joint is also applied to the adjacent surfaces of the stones, 
when these are not bearing surfaces, while bearing surfaces are termed 
beds . 

7. Modes of Bonding . It is important that the stones should be so 
arranged in any masonry structure as to tie the whole mass together 
laterally, and at the same time cause the various downward pressures 
to be evenly distributed toward and upon the foundation. This is accom- 
plished in part by avoiding continuous vertical joints fron course to 
course and, indeed, by separating or breaking such joints as widely as 
possible; and, farther, by the use of headers and stretchers . (Fig. 2) 
Stretchers are placed with their longest dimension showing in the face of 
the wall; while headers (also called binders and throughs) have their 
longest dimension running transversely through the wall. 

Headers should be evenly distributed through the mass of masonry, 
and to advantage headers and stretchers may regularly alternate in each 
course, the headers in one course being arranged so far as practicable 
over the middle of stretchers in the course below, or at least so as to 
break joints by a foot or more. Rigid specifications generally require 
that a certain proportion of the face area of the wall, ranging from one- 
fifth to one-third, according to the character of the work, shall be 
formed by headers, and also fix the allowable maximum and minimum dimen- 
sions of both headers and stretchers. In figures 7-12 are shown 
various possible arrangements of headers and stretchers for an isolated 
wall; and in figures 13 - 21 are shown similar arrangements for the case 
of two walls meeting at an angle, and thus forming a corner or quoin . 
The plan usually adopted at a corner is to have a stretcher run out 
alternately into each wall in the successive courses as in figure 13. 
It is preferable, in order to tie the walls together in the best manner, 
to avoid joints at the intersection A B (Fig. 14) of the inner faces. 
It is also an advantage that courses shall be continuous through both 
walls, in order that there may be the same niamber of bed joints in each, 
and in consequence a uniform settlement throughout the structure. 

In special cases where a stronger bond is desired than that afforded 
by headers and stretchers alone, as in some light-houses, piers, dams and 
other works, further expedients are adopted, such as dovetailing, or 
otherwise interlocking the stones, or using iron dowels, cramps, tie 
rods, etc. (Fig. 3). 

In brick walls the bricks are laid variously with, 

(a) Flemish bond, each course made up of alternate headers and 
stretchers (Fig. 4) . 

(b) English bond, having alternate courses of headers and stretch- 
ers (Fig. 5), anA considered stronger than the Flemish bond. 

(c) A modification of the English bond in which a course of headers 
alternates with from four to six courses of stretchers (Fig. 6), 
the arrangement commonly employed in this country. 

(d) Other less common arrangements of bricks, such as diagonally 
across the wall. 

-5- 



8. Olassifieation of Masonry . Stone masonry is commonly classified 
according to the nicety with which the joint surfaces are prepared, and 
also according to the methods of arranging the stones in the structiire. 
In all but the most inferior work the stones are imbedded in hydraulic 
cement mortar, which assists in giving uniform distribution of pressure, 
unites the whole mass of stone work, and prevents the entrance of water 
and gases at the joints. 

Rubble Work is composed of stones of irregular shape, no especial 
care being taken to dress the bed or other joint suirfaces. The spaces 
between the larger stones must as a rule be filled in with smaller stones 
and mortar. If mortar is not used then we have dry rubble . Rubble is 
variously employed in retaining walls, as backing for the higher classes 
of masonry, for side walls of sewers, foundations of buildings, etc. 

Squared Stone Work is superior to rubble, both in strength and in 
appearance, and is composed of stones roughly squared, and dressed to 
give joints one-half inch or more thick. This class of masonry is large- 
ly used in heavy retaining walls, abutments, piers, and other massive' 
work. The distinction between it and ashlar is frequently ignored, both 
being included in the latter class. 

Ashlar is a general term applied to cut stone masonry, and in par- 
ticular to that composed of stones with joint surfaces dressed to give 
joints less than one-half inch thick. It is employed wherever the 
highest class of work is desired, as in piers, arches, canal locks, gate 
chambers, engine foundations, etc., and is often backed with irubble. If 
the individual stones have all their dimensions assigned in advance of 
cutting, they are termed dimension stones . 

Masonry may further be classified according to the extent to which 
the stones are arranged in continuous horizontal courses, as:- 

(a) Coursed or range work. 

(b) Broken range work. 

(c) Random work. 

In railroad specifications sundry arbitrary classifications of 
masonry are made, as for example into first, second and third class arch 
culvert; box culvert, etc. 

9, Selection of Stone. The stones most often employed in engineer- 
ing structures in this country are varieties of granite, sand-stone and 
lime-stone. Granite occurs in various shades of red and gray and can be 
given a high polish. It is hard and therefore expensive to cut, but is 
suited to stmctures requiring great strength. Its color, hardness, 
durability, susceptibility to polish, and liability to discoloration, are 
all largely determined by the structure of the feldspar contained. 

Seuidstone occurs in gray, brown, red, buff and blue colors, and does 
not take a polish. It consists of grains of sand, cemented together by 
either silica, carbonate of lime, or oxide of iron, these materials 
exerting a controlling influence upon the character of the stone. Silica 
renders the stone light colored and intensely hard, so that it can be 
worked only with great difficulty. Carbonate of lime causes it to be 
soft and crumbly, so that it easily disintegrates. Iron oxide gives a 
medium and satisfactory hardness, as shown in Portland brown-stone. 

Lime-stone, as used for engineering construction, is found in shades 
of gray and blue, and in general is easily worked. Those varieties sus- 

-6- 



ceptible of high polish are classed as marble, occur in many beautiful 
colors, and are limited in employment to ornamental construction. 

Building stones from different quarries vary widely in their dura- 
bility under exposure to variable climate and to the gaseous atmosphere 
of cities, and the behavior, \inder such conditions, of stone from a new 
quarry cannot generally be predicted with certainty. Nearly all sand- 
stones are to some extent permeable to moisture, and are hence liable 
to disintegration if exposed to freezing soon afttfr being quarried; but 
if allowed to "season" during the warmer months of the year, before 
exposure to the cold, they withstand freezing weather without difficulty, 
the quarry water having evaporated. Frequently such stones, and espec- 
ially limestones, possess the important advantage that when newly quar- 
ried they are soft and can be worked into any desired shape with perfect 
ease, while as they season they grow hard. 

10. Shaping the Stone . The rough blocks from which stones are to 
be cut are to be assumed as approximately rectangular in shape. The 
workman is furnished with suitable sketches, the necessary dimensions and, 
if the character of the work requires, with patterns and templets or 
even models. More or less verbal directions and explanations must also 
usually be given. It is said that perhaps the most common source of 
error in difficult stone-cutting is the act of the workman in pushing 
blindly ahead with his work without fully comprehending the true shape 
of the stone. Pig. 26 is a copy of a workman's "paper", as given out 
to him with the rough block, and illustrates the practice of a certain 
large granite quarry. There are, as a mle, different methods of pro- 
cedure at command in cutting the stone to shape, especially in difficult 
work, varying in any particular case as to convenience, the amoxint of 
labor required, and the probable accuracy of result. 

Commonly the first operation is to dress off some one plane face, 
which is accomplished by manning straight chisel drafts successively 
along the boi'ders, these drafts being brought to a true plane by apply- 
ing two straight-edges, of equal width and with parallel sides, to dif- 
ferent drafts and sighting across the edges (Pig. 27). The surface 
enclosed by these preliminary drafts can then be reduced to a plane by 
the proper tools, the workman testing from time to time by a straight- 
edge applied in different directions across the surface. The operation 
here outlined is teiroed taking out of wind and can next be extended to 
other plane faces, whether they are at right angles or not, by the aid 
of a steel square or the bevel , the latter having a movable blade so 
attached to a straight stock that any desired angle can be set between 
them. 

Any Cuirved surface having rectilinear elements may be obtained in 
stone-cutting by first chiselling drafts or channels to correspond to two 
or more directrices, and then cutting the finished surface with reference 
to these, being guided by a straight-edge applied from time to time to 
the directrices in the direction of rectilinear elements (Pig. 28). 
Channels may be cut at short intervals corresponding to such elements, 
and the intervening surfaces then reduced by eye. 

For warped surfaces, so-called twist rules are very consnonly eipployed 
(Pig, 29). These consist of one or more tapering straight-edges, so 
framed to a rule having parallel edges that when the upper edges of the 
rules are in a common plane the lower edges will lie in a warped surface 
such as that required. Channels are cut along the stone to receive the 

-7- 



straight-edges, and are sunk until, as shovm by the above test, the chan- 
nels have reached the required surface. These channels then serve as 
directrices, to which straight-edges may be applied for determining 
elements, as already indicated. 

In all this work, especially on stones of irregular shape or in- 
volving curved surfaces, the workman is aided in shaping the surfaces 
and in verifying them by the various patterns and templets. The drafts- 
man who is charged with the duty of preparing these makes a full-size 
drawing upon a floor or other suitable surface, as previously explained; 
decides what procedure shall be followed in cutting the stone; cuts such 
patterns as are needed to the outlines of the faces as shown in the 
df awing; afterwards trimming them so as to make the proper reduction for 
joints. They can then be turned over to the stone-cutter, who will work 
much more closely and uniformly to these patterns than he would to 
dimensions, Lejeune, whose plates are largely employed in these notes, 
opposes the use of bevels, as not giving sufficiently accurate results, 
but on account of their convenience, and the economy of labor which they 
permit, they are likely to be employed in practice, except perhaps in 
the more difficult work. 

11. pressing of Stone Surfaces . The selection of a particular 
style of finish for the face" of the stone is partly a matter of taste, 
and is partly governed by a consideration of the peculiar uses to which 
the structure is to be put. For the face of a large and massive wall the 
rock face is appropriate, while for bridge seats, lock walls, gate 
recesses, etc., where the projections of the rock face would be inad- 
missible, a smoother surface such as pointed, or patent -hammered, for 
example, is required. Whatever the style of dressing adopted, even if 
the face of the stone be left rough, as in rock-faced and quarry-faced 
work, the edges or arrises are in general worked to definite lines for 
precision in laying and to give a certain appearance of regularity to 
the work. 

The faces of stones used in rubble masonry are left about as they 
come from the quarry. The coarser ashlar or squared stone masonry may 
have the stones either: - 

(a) ftuarry-faced , the exposed faces. being left substantially as 
given at the quarry. 

(b) Pitch-faced , the stone being pitched away from the edges so 
that these remain clearly defined. 

(c) Drafted , a narrow chisel draft being carried along each edge 
of the face. 

Ashlar proper has the edges of the faces of the stones defined 
either by pitching or by chiseled drafts, while the main surface may 
be dressed according to any of a great variety of methods, some of the 
more prominent of which are described below, the style of finish taking 
its name usually from the tool by which it is directly obtained. 



-8- 



FOR GRANITE SURFACES :- 

(a) Rock face , the irregular quarry face of the stone being left, 
except as modified by pitching. This mode of dressing is now 
used in most cases where at all admissible. 

(b) Pointed , the surface being more or less roughly indented 
through the use of the point , with hansner or mallet. 

(c) Pean-hammered or Pean-axed . the surface being reduced with the 
pean-hammer or axe, a tool having two opposite cutting edges, 
and giving a very rough finish, commonly to be seen on granite 
sidewalk curbing. 

(d) Pat ent -hammered , similar to pean-hamnered, but much finer, the 
finish being obtained by the patent hammer, a double-headed tool 
containing two sets of chisels, which give at each blow a se- 
ries of parallel cuts, ranging from 4 to 12 to the inch, accord- 
ing to the fineness desired. The surface is correspondingly 
known as "4-cut, " "5-cut," and so on. The 6-cut finish is the 
variety of pat ent -hammered probably most often used in engi- 
neering work, and is frequently selected fcr grooves, water- 
ways, gate-frame seats, etc. 

Other tools commonly used in dressing granite are the set or pitching 
tool, used for dressing edges of a block to line; the spalling hammer , 
sometimes used for taking off larger projections than can be removed 
with the set; and various chisels , used for finishing moldings, cutting 
drafts around rock-faced and pointed work, dressing portions of surfaces 
not easily accessible with patent hammer, lettering and tracing. The set, 
point and chisels are driven with the hand-hammer. 

The ordinary steps in the process of dressing a giranite surface are 
as follows, these steps being successively pursued as far as necessary 
to give the desired finish, and the cost, of course, being increased by 
each operation: - 

(1) Dressing edges to line with pitching tool. 

(2) Roughing out surface with point. 

(3) Cutting down irregularities left by point with pean hammer. 
(4r) Dressing down with 4, 6, 8, 10 and 12-cut patent hammers, 

successively, the irregularities left by each preceding tool. 
Machinery has been applied to some extent to sawing, polishing, and 
otherwise dressing granite. Surfaces are prepared for polishing with the 
lO-cut or 12-cut patent hammer, or in some cases by turning in a special 
machine, the process of polishing consisting in rubbing first with sand, 
then with emery, and finally with putty powder. 

FOR LIMESTONE SURFACES :- 

(a) Rock - face . 

(b) Pointed . 

(c) Tooled , the surface showing straight, parallel ridges and in- 
dentations, made by the chisel. 

(d) Drove , a siirfaee somewhat similar to tooled, but showing wavy 
stripes, and also made with a chisel called the drove . 

(e) Rubbed , a smooth finish. 

Thickly bedded limestones are cormionly sawed into blocks with gang 
saws. Some thinly bedded stones have beds smooth enough to be used in 

-9- 



ordinary ooarse work without dressing, and require only the faces and 
vertical joints to be dressed. 

Marble is usually finished with either a drove, tooled, or polished 
surface. For carving this and other stones various gouges, chisels, 
drills and othex- special tools are employed. The steps in the process 
of cutting marble are:- 

(1) Shaping up the block with spalling hammer and pitching tool. 

(2) Roughing out surface with point. 

(3) Cutting down projections left by point with tooth - chisel . 

(4) Cutting surface smooth with drove. 

FOR SANDSTONE SURFACES: 

(a) Rook face . 

(b) Pointed . 

(c) Crandalled and Cross - crandalled , a somewhat regular and fine 
pointing given by the erandall , a tool having a handle about 
2 feet long and keyed into the end a series of double-headed 
steel points. 

(d) Tooled . 

( e ) Tooth - chiselled . 

(f) Bush-hammered, a fine pointed surface given with the bush - 
hammer , the head of which is a square prism of steel with the 
end faces cut into pyramidal points. 

(g) Rubbed , the stone being at once sawed to a true surface, and 
then rubbed with another piece of sandstone, sand and water 
being applied during the operation. This is the finest finish 
of which sandstone is susceptible, and is often to be seen on 
the fronts of buildings. 

12. Special Constructions . The following constmctions not always 
given in books on drav/ing may be of service :- 

(a) To construct an arc of a circle by points, having given the 
span and rise, that is to say, having given three points (See 
Breithof: Coupe des Pierres), 

(1) Griven A, B, C, (Fig. 30). Draw chords as shown and upon 
these lay off the equal lengths A D and C E. At D and £ 
erect perpendiculars on which lay off, above and below the 
chords, equal spaces. Draw a line from A +hrough the first 
division above D and one from C through the first division 
below E. These lines will meet at a point F on the re- 
quired arc, and other points can similarly be found. 

(2) Given A, B, C, (Fig, 31). From Aaiidc as centers, with 
equal radii, describe arcs as shown. Sub-divide m s into 
any number of equal parts, and continue the equal divis- 
ions upon the opposite side of the chord A B, Lay off the 
common arc m a also successively each way frcan n. Draw a 
line from A through the first division below m and one 
from C through the first division above n and these lines 
will meet at a point F on the required arc, other points 
being found in a similar manner. 

(b) To draw a normal to an ellipse at a given point. 

-10- 



(1) The bisectrix of the angle formed by lines from the foci 
of the ellipse to the point in question will be a normal 
(See Fig. 32). 

(2) Complete the rectangle C upon the semi-axes (See Pig. 33) 
and draw its diagonals. Draw p 1 perpendicular to A 0, 
meeting C at 1 and then draw 1 n perpendicular to A B 
meeting A at n. n p will be a normal. 

(c) To find the plane angle which measures the solid angle between 
any two plane faces:- Assume the solid cut by an auxiliary 
plane perpendicular to the edge of the solid angle in question. 
This plane will cut from the adjacent faces lines enclosing 
the required plane angle. Now conceive a perpendicular in 
space from the vertex of this plane angle to the H trace of 
the auxiliary plane. The line will be perpendicular not only 
to the trace but also to the edge of the solid angle. If the 
projecting plane of the latter line be now revolved into H the 
perpendicular will move with it and. the true length of the 
perpendicular will become evident. This having been deter- 
mined, the revolution of the vertex of the desired plane angle 
into H is a simple matter and the angle is then revealed in 
its true size. Thus in Fig. 34, where the various planes are 
given by their traces, draw t - t, the H trace of the axjxiliary 
cutting plane; revolve m into H as shown, and the line a b, 
pei^pendicular to the revolved position of m, will be the true 
distance in space from a point in the horizontal plane at a to 
the edge of the solid angle, c d e then represents the re- 
quired angle revolved into H, and therefore shown in its true 
size. The procedure would be similar for working in V. 

"13. Walls with Plane Sloping Surfaces . A wall having a sloping 
face is said to have a batter , and this is usually expressed in inches 
per foot, a batter of 1 inch to the foot, or 1 in 12, referring to a 
elope at the rate of 1 horizontal to 12 vertical. Such a wall is shown 
in Pig. 22, where the batter is the ratio of the offset P M to the corre- 
sponding height MB. A stone of the lower course in this wall may be 
cut as follows;- Let a rectangular block be assumed, of width at least 
equal to A B, of height A E, and of any suitable length. Let A B M E 
E'A'B'M* (Pig. 23) be such a block. First dress the lower bed A B B'A'; 
then, working from this surface with the square, dress the vertical face 
A E E'A', and the joints A B M E and A'B'M'E', which should be square not 
only with the lower bed but with the vertical end face. Next dress the 
upper bed E M M'E* square with the vertical faces and at the proper 
height above the lower bed. It remains only to cut the battered fa9e, 
which may be done by first laying off A B and A'B' equal to A B (Pig. 22), 
erecting peirpendiculars at B and B', then making M F and M'F' equal to 
M P (Pig. 22), drawing the lines B B' , B P, B'P', P P', and finally cut- 
ting away the triangular prism B B'M M'P P'. 

A preferable mode of working is as follows:- dress a vertical joint 
surface first, tracing upon it by means of a pattern the outline A B P E 
(Pig. 22), Then, working from this surface with the square, dress the 
end faces and the bed joints, finally cutting the other vertical joints 
parallel to the one first dressed. 

-11- 



If the batter is considerable, and it is desired to avoid acute 
angles between the bed joints and the battered face, this may be done 
by stopping the horizontal beds a few inches distant from the sloping 
face, and then carrying them out at right angles to that surface, as 
shown at G D H and SOP (Pig. 24), The bottom stone could be finished 
as shown by the profile P N M B. 

A stone of the bottom course csui in this case be cut as follov7s:- 
Choose a block having the height B I, the width A B (Pig. 24), and any 
suitable length. Let a b q p a'b'q'p' (Pig. 25) be such a block. Begin 
by dressing the face which is to fonn the lower bed, and upon this trace 
the rectamgle a b b'a', having the sides a b and a*b' equal to A B (Pig. 
24) and the sides a a' and b b' equal to the assTjmed- length. Next, by 
means of the square dress a plane face a b q p perpendicular to the lower 
bed and passing through the edge a b. Upon this face, by means of a 
pattern, trace the contour a e c f n m b, identical with AECPNMBof 
Pig. 24. Then, by aid of the square, cut all the faces that are perpen- 
dicular to a e c f n m b, giving to these faces the assigned length, which 
will peiTOit of easily cutting the remaining face a'e'c'f 'n'm'b' . 

If it is desired to work more accurately, one may cut not only 
a b q p but a'b'q'p' square with the lower bed, tracing upon a'b'q'p' 
the contour a'e'c'f 'n'm'b ' identical with a e c f n m b. Thus for each 
of the remaining faces the workman will have two directrices in the same 
plane, such as a e and a'e'; e o and e'c'; c f and o'f, etc., and the 
faces can be cut with ease and precision. 

It should be noticed that the broken joint E C P produces a salient 
angle in the second course and a re-entrant angle in the first course. 
It is somewhat difficult to cut the latter angle with accuracy, and it 
is also likely that the two angles will not be cut precisely alike, in 
which case there will be imperfect contact between the faces and a liabi- 
lity to fracture during settlement. Special care should, therefore, be 
taken with the cutting of broken joints. 

14. Walls with Cylindrical Surfaces . The first case to be consid- 
ered will be that of a wall whose two faces are circular cylinders with 
concentric bases. 

Let A B C D suid E G P H (Pig. 35) be concentric circles with a 
common centre at 0. The horizontal projection of the wall will be in 
the annular space between the two circxanferences. The height of the 
wall being assumed, the vertical projection is easily obtained as shown. 
Now divide the circumference A C B D into a certain number of equal 
parts, 8 for example, and through the points of division draw the lines 
A E, M N, C G, etc., all converging toward and forming the horizontal 
projection of the vertical joints of alternate courses, beginning with 
the upper. Then divide anew A C B D into 8 equal parts in such a way 
that the points of division shall fall midway between those first obtain- 
ed, and through the new points of division also draw right lines con- 
verging toward 0. These lines will be the horizontal projections of the 
vertical joints of courses alternating with those first mentioned. 
Finally, draw horizontals such as P Q at suitable intervals to represent 
the bed joints of the courses, and through the points of division of 
A C B D draw perpendiculars to the ground line to determin'e the vertical 
joints in the elevation. 

A stone such as that having for horizontal projection the figure 
M N G C and whose exterior face is vertically projected in M'M'C'C 

-12- 



oo\Ald be cut as follows:- Cut a pattern to the contotir M N G C, and 
choose a block of stone having fhe height M'M", and its other dimensions 
such that the pattern can be applied to either o:f the quarry beds p t ^ c' 
^Fig. 36). By means of the pattern trace upon one quarry bed (assumed to 
have been brought to a plane) the contour m'n'g'c'. Then through the 
points m',n',g , draw upon the corresponding faces of the stone the 
right lines m'm, n'n, g'g, perpendicular to the quarry bed, apply the 
pattern to the face q u r c so that the vertices shall fall at the 
points n, m, g, e, and trace the outline of the pattern. 

The vertical joints, which are plane faces, are easily cut, since 
for each of them we have four right lines upon which the straight-edge 
may be applied. The cylindrical surfaces m'c'c m and n'g'g n can be cut 
by working so that the straight-edge always rests upon the two circular 
arcs limiting the faces at top and bottom, maintaining the straight-edge 
parallel to itself in the successive positions. This can be accomplished 
by dividing the arcs ra o, m'c', for example, into the same number of 
equal parts c'x = o y, x v = y z, v m'= z m, and applying the straight- 
edge so that it shall occupy the positions x y, v z, etc. In other 
words, the straight-edge should always be directed along one of the 
elements of the cylinder. 

The mode of cutting above explained is disadvantageous in occasion- 
ing a considerable loss of stone, and is therefore inferior to the 
following:- Choose a block having the height of the course, and its 
other dimensions such that the pattern can be applied in the position 
m'n'g'c' (Tig. 37), trace the right lines m'm, n'n, g'g, c'c, perpendicu- 
lar to the quarry beds, and finish the stone as before with straight-edge 
and square. 

It is plain that a block of smaller dimensions than before used 
answers in this second method. The difference is shown in Pig. 38, 
where p t s c' represents the size of rectangle required in the first 
case, and t p's'c" that in the second case, evidently smaller than the 
other. 

15. Cylindrical Wall with Elliptical Base . If it is desired that 
the interior face of the wall be a cylinder having for its base an ellipse 
similar to that which deteiroines ths exterior face, an ellipse will be 
drawn having its axes proportional to those of the exterior ellipse. The 
wall will in this case evidently be thicker in the direction of the major 
axes than in that of the minor axes. The vertical joints c£in no longer, 
as in the case of a cylindricsil wall with circular base, converge towards 
the common centre of the two faces, since they would then be perpendicu- 
lar to neither of the ellipses, except in the line of the axes. Further, 
if the joints were made normal to one ellipse, they would not be normal 
to the other. This difficulty may be avoided by using broken joints, one 
part being normal to the exterior curve, and the other normal to the 
Interior. Broken joints have been stated (13) to be objectionable where 
pressure is brought to bear upon them; but for vertical joints, as in the 
present case, with simply contact, there is no disadvantage in them. If, 
however, the wall were in any way exposed to a thrust, bringing pressure 
upon the broken joints, it would be better to substitute for them plane 
joints normal to a mean ellipse situated between the other two. In this 
case the joints would be nearly normal to each surface. 

-13- 



If it is desired that the interior face of the wall have for a base 
a ouirve everywhere equidistant from the exterior ellipse, construct a 
suitable number at normals to that ellipse, upon each of which lay off a 
distance eqtial to the desired thickness of wall, 

The cutting of the stones for an elliptical wall will be prosecuted 
in precisely the same manner as for a circular wall, en^loying one or 
the other of the methods previously explained (14). 

16. Walls with Oonical Surfac es. If it were required to join two 
walls having the same batter, ~ in such a way as to avoid an angle at the 
Junction, it would be convenient to employ a conical surface in the fol- 
lowing manner:- Let Fig. 39 show the plan of two walls of the same bat- 
ter, with the right section of one of them. Join the exterior faces of 
these walls by a conical surface, of which the vertex will be horizon- 
tally projected at S, at the point where the two right lines A B and C D 
meet, and vertically projected at the point of intersection of A'E' and 
C'P', a point lying, in this case, outside the paper. 

The conical surface will evidently be that of a riglrit cone with a 
circular bass, which will be cut by the planes of the bed joints of the 
courses along portions of horizontal circles, of which the centers, 
situated upon the axis of the cone, will all be horizontally projected 
at the same point S, which is, therefore, the horizontal projection of ^ 
the vertex of the cone. All these circles will be concentric in hori- 
zontal projection. 

The vertical joints will be indicated as shown in the drawing, tak- 
ing care to make them alteimate in the successive courses. If it is 
determined entirely to avoid the acute angles which the bed joints of the 
stone make with the exterior face of the wall, then one will make the 
joints, for a short distance from the sloping surface, perpendicular to 
that surface, thus producing conical surfaces all having their vertices 
upon the axis of the cone which forros the exterior face of the wall. It 
is rare, howeve.r, that the batter of the wall is sufficient to make this 
constmzction advisable: It is clear that the interior face of the conical 
wall will be a right circular cylinder having for a radius S B, and tan- 
gent to the faces of the two walls. 

If it is proposed to cut the stone whose horizontal projection is 
limited by the figure G H N M (Fig. 39), choose a block of stone having a 
height between quarry beds equal to the height M*M" of the course. Having 
dressed the lower face of this block to a plame, trace upon it, by means 
of a pattern cut along G H N M, the contour g h n m (Pig. 40); through 
the points m and n draw perpendiculars to the bed, and to the upper bed 
apply a second pattern cut along I M N K, thus tracing the contour i m'n'k, 
Finally, draw i g and k h. The interior face, which is cylindrical, will 
be cut as has been directed for cylindrical walls (14); that is to say, 
the workman will be guided by a straight-edge applied to the arcs m n, 
m'n' in such a manner that in each position it is parallel to m m' and 
n n'. The exterior face, which is conical, will be cut by using, to 
verify the work, a straight-edge which will be made to pass through cor- 
responding points of division of the ares i k and g h, which preferably 
will have been divided into the same number of equal parts. The joint 
surfaces, which are planes, will be easily dressed. 

Just as was explained for cylindrical walls, there is here a second 
method of cutting, more economical in the use of stone. Fig. 41 will 
make clear the features of the second method. 

-14- 



17. Oblique Conical Wall . TOien two straight walls of different 
batter are to be joined by a curved wall, the surface of the latter nat- 
urally is made that of an oblique cone with a circular base. The same 
surface is also sometimes employed for the end of a bridge pier, to Join 
the slightly battered aide surfaces with the line of the cut-water. 

Let A B and A C (Fig. 42) be the horizontal traces, prolonged, of 
the battered faces of two walls which are to be joined and whose right 
sections are shown; D E and D P the horizontal projections of the upper 
edges of those faces, and G H and G I the horizontal traces of the verti- 
cal faces of the walls. Through A draw a right line A o dividing the 
angle CAB into two equal parts. Every point on A o will be equally 
distsmt from the sides of the angle; consequently, from points upon A o 
circles may be drawn tangent to both sides of the angle. Take then a 
point o, distant from the sides A B and A C by an amount equal to the 
radius desired for the base of the xjone, and the arc M N, having o for a 
denter, will be tangent to A B and A C. 

Similarly, bisect the angle E D F, and upon the line D 04 thus ob- 
tained take a point 04 distant from the sides D E and D F by an amount 
equal to the radius desired for the arc which is to join the upper edges 
of the sloping faces. From 04 as a center describe the arc P Q, and draw 
the right lines M P and N Q. These lines meet at a point S which is the 
horizontal projection of the apex of the oblique cone which is to join 
the two sloping faces and which will be tangent to them. If now we draw 
a right line through o and 04 it will pass through S, since in an oblique 
cone with a circular base all plane sections parallel to the base are 
circles having their centres upon a straight line passing through the 
apex, o S is the horizontal projection of this line. 

It is evident that the circles which are to connect the Joints sep- 
arating the courses will have their centres upon o S, between o and 04, 
at intervals proportional to the heights of the courses. The vertical 
joints of the conical portion of the wall could bt9 determined by vertical 
planes perpendicular to the trace of the cone; but it is preferable to 
make them perpendicular to a horizontal section at mid-height of each 
course. 

In order to cut one of the stones of ^he conical wall, that for 
example horizontally projected in m n q t r p (Fig. 42) we may proceed 
as follows:- Dress the two bed joints and with a pat t era trace upon the 
lower the contour m n t r (Fig. 43) and continue as explained in No. 16. 
In dressing the conical face mark upon the stone the points where the 
edges p q, q n, n m, and m p are crossed by a suitable number of elements 
of the cone, working accurately along the lines thus determined and dress- 
ing intervening portions of the -surface by eye. 

18. The Flat Arch or Plate-band . This is a term applied to an 
Arrangement of wedge-shaped stones having a plane soffit and spanning a 
door, window or other opening. The sides of the opening are called the 
Jambs . The division of the arch into separate stones .is effected as 
follows:- Upon the ri^t line A B (Fig. 45) which represents the verti- 
cal trace of the plane forming the top of the opening construct an equi- 
lateral triangle A B; then through the vertex draw the lines F E, 
H G, OKI, etc., cutting the line A B at points F, H., K, etc. which 
divide that line into an odd number of equal parts A F = F H.= H K, etc, 

-15- 



the nvanber of divisions being decided by judgment. 

Experience has shown that when there is a tendency for the flat 
arch to fail vmder the load which it has to support it is by a rotation 
resulting in the opening, underneath, of the joints on either side of 
the key; and, above, of the joints near the jambs, as illustrated in 
Pig. 46. To avoid such results various expedients are adopted, such as 
increasing the thickness at the key, diverting pressure by means of a 
relieving arch built above the flat arch, tying together the stones of 
the latter by an iron rod, etc. 

The slipping of the extreme stones is sometimes prevented by break 
ing the outer joint along a horizontal plane a b (Pig, 45). The same 
arrangement may be continued toward the centre, taking care not to give 
too great length to the horizontal part of the joint, so that it may 
better resist tanequal pressure resulting from settlement. It is thought 
best, however, to leave th& key stone with unbroken joints, that it may 
settle firmly into place, with perfect contact on either side. 

A somewhat different arrangement is sometimes employed in which the 
joints are broken as along the line f g h n (Fig. 45), each stone catch- 
ing upon the adjacent one by a short offset g h n. As the appearance is 
not entirely pleasing to the eye, the offset may be made for only a 
part of the thickness of the flat arch, thus giving a projection and 
indent as shown in figures 47 and 48. 

In order to avoid the disadvantage of giving the stones acute an- 
gles at the intrados, vertical cuts such as x y may be made at the points 
of division of A B (Pig. 45), meeting a horizontal line p q distant a 
short way from A B, the joints being continued beyond from the direction 
of 0. 

In Pig. 49 let u v and x y be the horizontal traces of the faces of 
a right wall to be pierced by a flat arch; A A" C C E the horizontal 
trace or outline proposed for the side of the opening; A'B' the vertical 
projection of one-half the intrados, and A'P' that of the face of the 
Jamb. Draw G'D' and C'K' respectively parallel to A'B' and A'P*, and 
distant from them by an amount equal to C A". Above E erect the vertical 
V'E', prolonging it to a point E" situated above G'D' at a distance 
arbitrarily chosen, emd through E" draw a parallel to A'B'. Then divide 
the entire line A'B' representing the intrados into a suitable odd number 
of equal parts, and through the points of division erect the verticals 
A'L', M'G , etc., afterward drawing the lines L'T', G'Q', etc., all 
radiating from 0, a point determined as previously explained. Through 
T' draw a horizontal T'S', and by projection determine in plan the lines 
A L J, M G H, etc. 

An arch stone, such as that vertically projected in R'Q'G'M'A'L'T'S' 
(Pig. 49) may be cut as follows:- Starting with a stone of a length 
equal at least to the thickness v y of the wall, and with the other two 
dimensions sufficient to contain the pattern of the head R'q'G'M'A'L'T'S', 
dress the joint surface u y y'n' (Fig. 50); then squaring from this face 
cut the two faces u v x y and u'v'x'y' (Pig. 50). Now, upon one of these 
faces apply a pattern of the head and trace the contour r, q d c f b t s, 
after which draw through the points r, q, c, s, which are upon the edges 
of the face u v x y, right lines r r", q q', cm', s s', which meet the 
edges of the other face at points r' , q', m'', s', which serve as guides 
for applying a pattern and tracing the contour r'q'g'm'a'l' t 's' . The' 
stone will now be cut as though it were to be a prism having for bases 

16- 



the contours above determined by the pattezms. This prism having been 
out, lay off the distances a'a and m'ln, drav a m, and through a and in 
draw, square with the edges of the Intrados, the lines a 1* and m g'; 
then lay off 1 1* and g g* and join 1 with g; next draw h J parallel to 
e f at a distance from it equal to B* z (Pig. 49); and finally join j with 
1 and h with g. It remains then simply to cut the three faces h j 1 g, 
g 1 I'g*, and g'l'a m which form the onbrasure. 

In order to out a skewback stone, begin as though there were no em- 
brasure, thus obtaining a prism of which the bases u x a 1 t and u'x'a' 
I't' are determined by means of a pattern cut along the contour U'X'A'L'T' 
(Fig. 49). That accomplished, trace upon the bed joint of the stone the 
contour x e'y'y a'a'x' by means of a pattern cut to the horizontal pro- 
jection of the skewback, that is to say, to the contour X E C"C A'A U 
(Pig. 49). Through e* (Pig. 51) draw the right line e*e square with the 
edge a x; make e e' equal to E'E" (Pig. 49); through e (Pig. 51) draw e j 
parallel to a x; through a" draw a"n square with a'a"; lay off n m, and 
through j and m draw j m. The skewback is then entirely outlined and 
the cutting is easily accomplished. 

If the flat arch is in a wall with battered face, begin by cutting 
the stone as though it were in a right wall, and then give the face the 
proper batter as indicated in connection with Pig. 23. If the flat 
arch is in a cylindrical or conical wall, again proceed as though it 
were in a right wall, afterward cutting to the proper curvature as has 
been explained in Sees. 14, 16 and 17. 

The Flat Vault ( French . VoSte Plate ) is a vault having for its 
intrados a plane surface, and therefore is somewhat emalagous to the flat 
arch. This construction will not here be considered in detail, but is 
illustrated in Pigs. 52 - 57, 

19. Wing Walls ;- The abutment of a dam or bridge is usually flanked 
by a v/ing wall, serving in part as a retaining wall to support the em- 
bankment behind it and in part as a protection of the latter from scour. 
The height of the wing wall usually decreases toward the end, in the case 
of a bridge abutment at least, either in successive steps or by a ixniform 
slope given to its top, and the thickness at the base diminishes accord- 
ingly. The stones forming the steps of the slope are continued well under 
those of the overlying course, and v/hen the slope is covered with a coping 
it is often finished at the foot by a heavy newel stone. Wing walls take 
a great variety of shapes in plan and meet the abutment at various angles, 
depending upon the conditions peculiar to each case. Preceding plates 
will serve to illustrate this construction, which is arbitrary as to 
various details. 

20. Pyramidal Buttress ;- It is sometimes desired to reinforce a 
wall by buttresses projecting from its facel Let it be required to make 
the drawings for a buttress of which the transverse section, the outline 
of the base, and, the batter of the faces are assumed to be given. 

The right section of the buttress and of the wall against which 
it rests, and the outline R M K V (Pig. 64) of the base of the buttress, 
can be drawn at once from the given dimensions. It remains to find the 
Intersections of the various faces. To determine the direction of M N, 

-17- 



Imagine the adjacent faces of the buttress, produced if necessary, out at 
any convenient height by an auxiliary plane along a b and o d. These 
lines are respectively parallel to R M and M K at distances e f and g h 
determined by the height of the auxiliary plane and the batter of the 
faces. V. N passes through the intersection of a b and o d. The direction 
of R S is similarly foiond from the intersection of a b and t u. The 
positions of N and S P with reference to M K are given by the right 
section, and the main lines of the drawing are then fully determined. 

The manner of drawing the joint lines is easily seen. The line P V 
is divided proportionally to the height of the courses, and through the 
points of division the horizontal joint lines are drawn parallel to the 
outline of the base. The vertical joint lines show in plan at right an- 
gles to the horizontal joints* In elevation they appear Inclined on the 
side face of the buttress and should be made parallel to some well deter- 
mined g^ide line, such as P'y*. 

The uppermost stone of the buttress is shown in isometric view 
(Pig. 65), and can be cut from a rough block as follows:- First dress 
the lower bed A^ B^ Cj^ D^ and trace its outline by the pattern. Next cut 
the front and rear faces by using the bevel from the base already cut, and 
by patterns trace the outlines Ai D^^ Pi S, and Bj Cj Oi %. Two lines of 
the upper face are now determined and the face can be cut and, by applying 
a pattern, its entire outline ean be traced. Finally dress off the lat- 
eral faces Ai B^ Nj S-^ and Di G^ 0^ Pi, the bounding edges of which have 
all been determined by the other faces. 



lUIASONRY ARCHES. 

21. An arch is a curved structure having a cylindrical intrados, 
and ordinarily forming the roof over an opening. If the arch is of stone 
it is composed of wedge-shaped blocks, which are sustained in position by 
their mutual pressvire. If the length of the opening in the direction of 
the Eixis is considerable, as when it is more than a mere opening through 
a wall, the arch is in architecture classified as a vault , although this 
distinction is not commonly observed in engineering. 

Arches are classified according to various features, and especially 
as to the shape of a right section of the intrados, in this way giving 
rise to the terms semi-circular arch, elliptical, etc. 

The semi-circular arch, also called Round and Full - centered , has its 
intrados a semi-circle in right section. A right section of the Segmental 
arch shows a segment less than a semi-circle. These are the most common 
forms, the semi-circular being considered the more perfect type as to 
appearance, but weaker than the segmental and requiring greater vertical 
space, according to the length of span. The segmental arch exerts greater 
thmist than the semi-circular upon pier or abutment, but is itself strong, 
may be proportioned to occupy but moderate vertical space, and is the 
form usually selected for large spans. 

If the rise of the arch is less than the half -span, as in the case 
of an elliptical arch with minor axis vertical, the arch is sometimes 
described as svirbased ; if the reverse is true, as in the case of an ellip- 
tical arch with major axis vertical, the arch is said to be svirmounted . 
If the elements of the intrados are horizontal, the case is that oi a 

-18- 



horizontal arch; but if they are inclined to the horizontal plane the 
arch is termed a descent . 

As has been said, the voussoirs of an arch are wedge-shaped, the 
Joints radiating from the centre or centres of curvature of the intrados, 
this direction making them also normal, or approximately so, to the line 
of pressure in the arch. Although intrados and extrados are often de- 
signed as concentric surfaces, especially for small spans, they are by no 
means always so, for the conditions of stability may require an increased 
thickness of the arch toward the springing line, and in any event the 
extrados is likely to be left rough in constjru.otlon, except in the case 
of the ring stones* 

In French practice the extrados is often arranged as follows: After 
having fixed the thickness B b (Pig. 63) to be given at the crown, which 
thickness veu^ies with the span, the weight to be supported and the mater- 
ial of the voussoirs, and is governed by recognized rules and precedents, 
take upon the vertical B a length B 0* equal to from two- thirds to 
three- fourths the span A C; then with 0' b as a radius describe the arc 
a b c, which defines the extrados. 

The precise shape of the voussoirs, as well as the precise number 
into which the arch shall be divided, are largely matters of taste. In 
general the voussoirs should be someiriiat deeper (radially) than they are 
wide, smd in order to insure a key stone at the centre, there must be azi 
odd number of them. Their width as measured on the intrados is sometimes 
increased moderately from the key toward the springing lines. Sometimes 
the joints of the arch stones and the coursing of the adjacent exterior 
masonry are planned with reference to each other, and sometimes independ- 
ently. Most often each voussoir is terminated by a horizontal face P Q 
(Fig. 6£) and a vertical face ft R, the joints of the voussoirs being ad- 
Justed in some degree to the coursing Joints of the wall in which the 
arch is built. Sometimes the Joints are arranged as in the left-hand part 
of Fig. 62 , but this is objectionable. The various patterns and- teitq)lets 
required in cutting the voussoirs are to be made from full-size drawings, 
but often it is also advisable, especially for large and importeuit 
structures, to compute the dimensions. 

Arches of brick are of very coimion occurrence, being employed in 
aqueducts, sewers, buildings, and even sometimes for bridges. of consider- 
able spans. In engineering work the bricks are as a rule arranged in 
independent concentric rings. Joined to one another by mortar only, and 
giving the so-called Rowlock bond. In ornamental construction, as in the 
fronts of buildings, the bricks are frequently laid in the Header and 
Stretcher bond, the Joints being continuous from intrados to extrados. 
Such an arrangement involves either "rubbing" the bricks to a wedge-shape, 
or permitting wide joints at the extrados. The Block and Course bond and 
other devices have been employed to some extent in engineering structures 
with a view to breaking the continuous concentric Joints of the rowlock 
bond, and by grouping the bricks in the shape of stone voussoirs to cause 
the pressure to be more regularly distributed through the Diass of brick- 
work. 

/^^/ / /Onv/ock ^ 

Seader and. 

Stretcfijer- 



-19- 






Block and. 
Coia'se 



The following are common technical terms applied to the various parts 
of the arch:- 

Intrados or Soffit . The Inner surface of the arch. 
The outer surface of the arch. 



Extrados or Back . 

^The highest part of the arch. 

Line . The line in which the Intrados meets pier or abut- 
e perpendicular distance between springing lines. (ment. 



Crown . 

Springii 

Span . 

Rise . The height of intrados at crown above level of springing linea 

Voussoirs or Arch stones . The stones composing the arch. 

Ring Stones . Stones showing in the face of the arch. 

Key Stone . The middle ring stone. 

Arch Sheeting . The material forming the lining of the arch, espec- 
ially that portion not showing in the faces. 

Springer . The voussoir nearest the springing line. 

Skewbacf . The inclined surface of the stone from which a segmental 
arch springs. The term is also sometimes ap- 
plied to the stone itself. ^ 

Haunch . The portion of the arch midway between crown and springing 

Spandrel . The lateral space outside the extrados. (line. 

Course . A row of voussoirs running in the direction of the axis of 
the arch. 

Coursing Joint . A joint separating adjacent courses. 

Heading Joint . A Joint running transverse to the courses. 




22. Right Arch . This term is applied to a horizontal arch whose 
faces are perpendicular to the axis. In Fig. 60 is shown the elevation 
of such an aroh, with voussoirs arranged in accordance with principles 
already stated. 

In order to cut the voussoir projected in M N P- D C one may proceed 
as follows:- Choose a block of stone having a length at least equal to 



-20- 



the thickness of the wall (if the opening Is not too long to be covered by 
a single length of stone; otherwise choose any suitable length, depending 
upon oircTimstanoes) , and whose base x u d t (Fig. 61) can contain a pat- 
tern of the head M C D P N (Fig. 60). Dress the face x u d t and by 
means of a pattern of the face of the voussoir trace the contour m c d p q 
preferably arranging that the edge of one of the joints shall coincide 
with the quarry bed u v d'd. Now cut the joint surface d p p'd' passing 
through the right line d p square with the face of the voussoir, and out- 
line the contOTir of the Joint surface D P by constructing the right angles 
p'p d and p d d', of which the edges p p' and d d* are equal to the 
length of the stone. Proceed similarly for the joint surface M C, Then 
dress the second face of the voussoir, m'c'd'p'n', which will be easy, 
since two right lines p'd* and m'c' contained in its plane are known, and 
the work can be verified by squaring from the joint surfaces already out. 
The exact outline will be obtained by applying the pattern M C D P N 
(Fig. 6 0) so that M, C, D and P shall fall at their appropriate positions. 
The faces a p p'n* and n m m'n' can now easily be cut, since for each of 
them three right lines are known, to which the straight-edge can be ap- 
plied; they should further be verified with the square, applied to the 
end faces of the voussoir. The cylindrical intrados can be cut by apply- 
ing a straight-edge to corresponding reference points spaced at equal 
intervals on the end curves. 

23, A Horizontal Arch , Terminated at one end b^ a Sloping Skew 
Face , and at the other b£ another Arch of Larger Radius ; - 

The axes of the arches are supposed to be at right angles to each 
other, and their springing lines to lie in the same plane, which will be 
taken as the horizontal plane of projection. Both arches are here assiiai- 
ed to be semi -circular, and the larger to be constructed of brick masonry, 
not therefore requiring an adjustment to its courses of the voussoirs of 
the other arch, as would be the case if both were of stone. It is plain 
that the smaller arch will meet the larger along a curve of double curv- 
ature, and the sloping face along an ellipse, and it is between these 
curves that the various voussoirs to be planned will be comprised. 

The vertical plane of projection will be taken perpendicular to the 
axis of the smaller arch. Upon this plane trace the semi-circle F'S'E* 
(Fig. 66) a right section of the intrados of the smaller arch, and upon 
the horizontal plane the right lines DFEC, D*F"E"C" which will be, 
respectively, the trace of the sloping face and a springing line of the 
larger arch. Divide F'G'B* into an odd number ef equal parts, five for 
example, and through the points of division and the axis 0* 0" pass 
planes M'0'0", N'O'O", etc., which will form the joints of the voussoirs 
and will cut the cylindrical intrados along elements. Limit the joints 
M*JR', N*P*, etc., by a circle D'R'O* and terminate each voussoir by a 
horizontal and a vertical face such as R*(l' and Q*P'. 

One of the voussoirs will be vertically projected in the polygon 
M'N'P'q'R' and will occupy in the right prism which has this polygon for 
a base the space between the slope F £ and the intrados of the larger 
arch springing from P*E". It remains to find two things: Ist, the inter- 
section of the faces of the prism with the sloping face; and 2d, the 
intersection of the same faces with the intrados of the larger arch. 

In order to find the first of these intersections. Imagine the slop- 
ing face cut at D by a vertical plane perpendicular to D E; it is thus 

-21- 



intersected along a line horizontally projected in D A and which will make 
with a vertical at D an angle which measures the inclination of the face. 
Let Z D*A' represent this angle, here assigned in advance, turned and 
projected upon the vertical plane of projection. We may now find the 
horizontal projection of any point lying on the sloping face, the point 
vertically projected in M* for example, by conceiving a horizontal line 
passing through the point and lying in the sloping face, meeting the line 
D'A* at a point m* horizontally projected upon D A" at m. This point 
is then to be revolved into the plane D A by means of the arc ra m", and 
then through the point m", thus obtained, draw m*M parallel to D C, 
meeting the vertical dropped from M* at a point M, iriiich will be the re- 
quired horizontal projection. The other necessary points may be found in 
the same manner, after which draw the right lines MR, R ft, q P, P N, 
forming the polygon M N P ft R, which will be the horizontal projection of 
one end of the voussoir. It is to be noticed that the right line R ft 
ought to be parallel to D 0, since the face which has R Q-fbr £in edge is 
horizontal; also that the face which is vertically projected in P'ft', 
being, perpendicular to the vertical plane, will have all its edges hori- 
zontally projected upon the same right line perpendicular to the grotind 
line. 

The other face of the voussoir, that formed upon the intrados of the 
large arch, will be obtained in a manner similar to the above. That is, 
draw in the vertical plane an arc of a circle C*Y corresponding to the 
CTurvature of the larger arch; then through the different points such as 
N', ft*, etc., draw the horizontals N'n*, ft'q', meeting C'Y at points n', 
q', horizontally projected at n and q; then by means of arcs revolve these 
points into the vertical plane CO" and draw through the revolved points 
parallels to D"0", thus obtaining the desired horizontal projections 
N and ft. Other points may be found in the same manner. 

The joint surfaces meet the intrados of the large arch along curves 
which are portions of ellipses. The edges of the joints furnish certain 
points upon these curves, but others may be obtained, if needed, directly 
by projection, as illustrated for the point X. Since all the joint 
planes pass through the axis 0" of the smaller arch, it is plain that 
all the right lines such as M R and N P should converge to the point 0; 
the elliptical arcs N'R", N"P", etc., should for the same reason pass 
through 0", where they ought, further, to be tangent to D'C". 

24. Obtaining the Patterns:- The arch has now been determined, 
since the projections of the faces of the voussoirs have been ^ound. But, 
for the purpose of cutting the stones, it is further necessary to know sacth 
of the faces in its tru.e size, at least the intradosal and joint faces. 
Now all the intradosal faces are parts of the same cylindrical surface, 
and if this be developed they will all bo known in their true size. To 
this end, first rectify the curve of right section by laying off upon the 
indefinite riglrit line e f (Pig. 67) distances en, n m, etc., equal to 
the lengths of the arcs E'N', N*M' (Pig. 66). This may be done by laying 
off with the spacers a short chord the proper number of times, or more 
accurately by laying off the computed length of the semi-circumference 
and dividing it into the proper ntamber of equal parts. At the various 
points of division lay off indefinite perpendiculars from e f, upon these 

take e e'= I E, n n'= H N, m m'= KM f f'= L P, and through the 

points e', n', m', ... f, thus obtained, draw the curve e'n'm'f, which 
will be the development of the arc projected in P M N E. Points upon this 
curve intermediate between those above mentioned may be fovind by taking 

-22- 



elements of the cylindrical intrados lying between the edges of the 
voussoirs, locating these on the development and laying off the proper 
distances upon them. Just as vas done for n', m*, etc. 

In a manner similar to the above lay off the distances e e*= I'E", 
n n"= H N", m m"= KM" .... f f«= L F», and through the points e", n", m" 
.... f" thus obtained pass a curve 8'*n"m"f", which will be the develop- 
ment of the line of intersection of the two arches, horizontally project- 
ed in E'NTifl"?" (Pig. 66). The true form and size of all the intradosal 
surfaces are now known, and are irepresented by the curvilinear figures 
e'n'n"e", n'm'm'n", etc. 

In order to obtain patterns of the plane Joint siirfaces, lay off 
upon e f distances n p = N'P*, m r = M*R', etc., erect through the points 
p, r, etc., thus obtained, perpendiculars, upon which take p p*= W P. 
p p"= W P", r r'= T R, r r"= T R", etc., and the right lines n'p', m*r*, 
etc., will be the outer edges of the Joints. To obtain the opposite 
curved edges, find for each of them at least one point between n" and p", 
m» and r", etc., which may be done by applying the above method to hori- 
zontal right lines passing through the middle of the Joints N*P*, M'R', 
etc. (Fig. 66). 

25. Cutting the Voussoirs :- For cutting the voussoirs two distinct 
methods are applicable, known respectively as the method by squaring and 
the method by bevels . The method by squaring has the advantage of offer- 
ing the greater precision, but both methods will be explained by supposing 
them applied to the voussoir vertically projected in M'N'P'Q'R' (Pig. 66). 

Method by Squaring :- Choose a block of stone having a length at 
least equal to the greatest dimension ^"v (Pig. 66) of the horizontal pro- 
jection of the voussoir, and with its other dimensions such that a pattern 
of the face M'N'P'Q'R' can be applied upon the two end faces. 

Begin by dressing one of the ends to a true plane surface, and trace 
upon it the contour mi Nj p-a qi r^ (Pig. 68) by means of a pattern cut 
to the vertical projection m'N*P'Q'R' (Fig. 66), preferably turning this 
pattern so that the side M'R* of one of the Joints shall coincide with 
the quarry bed mx r-j^ Rg Mg. Then, by means of the square, cut a plane 
surface from m^ r^ exactly perpendicular to the end already cut, and 
apply the pattern m'r'r"m" (Pig. 67) of the upper joint, placing it at 
distances m^ M, = M u, r^ R^ = R z (Pig. 66), u N being parallel to the 
ground line, and thus outline t'he contour R^ M^ Mg R2 o^ the Joint. Pro- 
ceed similarly for the joint surface whose edge is Ni P^, applying the 
coi'responding pattern n'plp"n" (Pig. 67) and tracing the contour P^ N^ 
N2 P2 o^ the lower joint. 

The intrados is a portion of a cylinder passing through the curve 
m^ N^ and having its elements perpendicular to the base m\ Ni p^ q-^ r^. 
The square may answer for cutting this cylinder; but if too short for the 
purpose a templet may be employed, cut to the arc M'N' and moved upon the 
two right lines m^ Mo, n, Ng, already traced, in such a maxmer as always 
to be parallel to the plane of the arc mi N,. When the intrados has been 
cut, apply its pattern m'n'n"m" (Pig. 67) which should be of flexible 
material- in order to allow of close contact with the concave surface, 
and trace the end curves M^ N^ and Mg No. 

The lateral face q, p., Pg Qg and the uppeP face q^^ r^^ Rg Qg, being 
planes perpendicular to the base of the prism, and passing through two 
known right lines, may easily be cut by the aid of a straight-edge. This 
done, lay off the distances q^ q,i and q^ Qg respectively equal to v Q and 

-23- 



I 



V Q" (Pig. 66), which will permit of tracing the right lines R^ ft^ and 
(Ji P-i , and also the arc fto^Po t>y means of a templet cut to the curvature 
of tEe right section C'Y fPig. 66). 

It remains only to cut that face of the voussoir which lies in the 
intrados of the larger arch, and of which we already know the contour 
M2 N2 P2 Qs ^2* This face is a cylindrical surface and can therefore be 
cut by moving a straight-edge upon the contour M^ % ^2 ^ ^2 ^^ such a 
manner that it passes at the same time through marks a and b, a* and b', 
Mg and b'", etc., corresponding to elements of the cylinder. These 
marks can easily be determined by tracing upon the base m^^ n^ P]^ Qj Ri 
right lines parallel to q, ri , such as aj bi, and bringing forward the 
points a^^ and bj^ to a' ana b"* by means of right lines a-, a* and b^ b* 
parallel to the edges of the voussoir. The opposite end of the voussoir, 
being entix*ely a plane surface, with its exact contour known, can easily 
be out by the aid of a straight-edge, cutting away the excess of stone 
between the face and the base m, N^^ p^ Qi r, , 

Modified Method :- The method above described can be modified as 
follows, with the attainment, perhaps, of greater accuracy, and with 
avoidance of the necessity for using a variety of patterns:- Record 
upon a rough sketch the lengths of the different right lines M u, M'u, 
R z, R"z, P V, P'v, etc., and of any intermediate ones desired. Then 
having squared a right prism having for its base the contour m-^ Ni pi qi 
r^ (li'ig. 68), mark upon the stone the distances rj^ R^^, r, Rg, m, Mj^, m^ 1^ 
etc., as recorded upon the sketch; also trace the curves Mg Rgj^lg Ng, 
etc., of which one or two intermediate points should be known. 

Method by Bevels ;- After having dressed a quarry bed and having . 
traced upon it the contour M, Mg Rg Ri (Fig. 68) of the upper bed joint, 
cut an indefinite plane surface passing through M^ Mg and making with 
the joint already cut an angle equal to R'M'N (Fig. 66). This will 
serve provisionally as a substitute for the tirue intrados, and the proper 
angle will be obtained in cutting by the use of the bevel, held always in 
a plane at right angles to the edge M]^ Mg. Having dressed the indefinite 
plane, trace upon it the rectilinear contour M, Mg Ng Ni easily determined 
from the drawing, the plane would cut the intrados of the large arch 
along a curve, and not along the right line VL Ng, but that is of no con- 
sequence, since the plane is to serve but a temporary purpose. 

In the same way, by means of a bevel forming an angle equal to 
M'N'P' (in this case equal also to N'M'R'),. one of its sides being kept 
upon the flat intrados and its plane being always maintained perpendicular 
to the edge N^^ Ng, cut the lower bed joint, upon which trace the contoia? 
Nj Ng Pg Pi, using for this purpose a pattern of the joint. 

It will then be easy to dress the face which shows upon the battered 
surface, since it is a plane passing through the three right lines Rj. ^1* 
Ml Ni, N-, P^, which are known, and by means of a pattern the contour 
% ^1 Pi fti ^ '^^^ ^^ traced. This having been done, and the arc Mj. Nj 
being then known, cut out the cyliildrical intrados by using a templet cut 
to the curve M'N' (Fig. 66). Applying now to this cylindrical surface 
the pattern of the developed intrados, trace the arc Mg Ng which limits 
the intrados at the end. 

It remains only to cut the two plane faces Ri ft^ Qg Kg and Pj^ Q^ 
(ig Pg "and the curved end of the voussoir. The plane faces are easily cut. 



-24- 



since for each of them two right line directrices are known. Upon the 
edges R, Rg and q,-^ Qo °^ ^^^ first of these faces take R, Ro = R R* and 
ftl fto = Q Q', and join R2 and Qg by a right line, which ought to be per- 
pendlcixlar to the edges. Similarly, upon the edges ft-, Q2 ^^^ ^1 ^2 0^ *^® 
second face take ^1^2"^ ^" ^^^ ^1 ^2 ~ ^ ^"» ^'^^ through Q2 ^^^ **2 
pass a cur\'-e, to. be traced by means of a templet cut to the curvature of 
a right section of the larger arch. The remaining end face of the vous- 
Boir, a cylindrical surface of which the contour Mg Ng P2 Qg % ^^ com- 
pletely determined, can be cut by means of a straight-edge kept always 
parallel to the edge Rg ^* 

26, Remarks ;- Tne aisadvaiitage of having stones with acut6 angles 
has already been noticed. Now it will be seen in Fig. 66 that the hori- 
zontal surfaces of the voussoirs form with their end faces lying in the 
large arch angles which are the more acute as the voussoir in question is 
nearer the key and as the radii of the tv/o arches approach each other. 
This difficulty can be met by terminating these horizontal faces by a 
short face normal to the large arch. 

It is to be noticed also that the angle E C C" which the vertical 
face n 0" of the first voussoir forms with the slope Is the more acute 
as the obliquity of the wall increases. This voussoir mignt therefore 
be given a narrow face perpendicular to F. C, The same thing might be 
done for the other voussoirs at P, D, etc, although it would be plainly 
objectionable to do it at F. In case the skew were considerable resort 
would sometimes be had to the elbow arch, composed of two arches, one 
perpendicular to D C and the other to D''C". 

If the large arch were of cut stone, then the voussoirs in the 
smaller arch should conform to the courses of the larger, euid would be- 
come somewhat complicated, each voussoir at the intersection having two 
intradosal surfaces, one for each arch. A different drawing would then 
be required, and the problem would come undei' the head of Lunettes, to 
be considered later, 

27. An Arched Opening in a Cylindrical Wall or Round Tower : - 

Let the circular arcs A e B and C f D (Fig, ^) be the horizontal 
traces of the cylindrical wall or round tower, and let it be required to 
construct in this wall an arched opening to be comprised between two 
parallel planes A C and B D which are perpendicular to the chord C D of 
the arc intercepted by the opening. The intrados of the arch will be 
projected vertically along a cui^e A'U'B', here tsLken as a semi-circle. 
Divide this curve into an odd number of equal parts, five for exaqiple, 
and through the points of division and through the axis O'f of the cylin- 
der of the intrados pass planes M'O'f, N'O'f, etc, which will divide off 
the voussoirs and will cut the Intrados along rectilinear elements. 
Terminate the joints thus formed, M'R*, N'P', etc, by a circle E'R'F* 
concentric with the first, and limit each voussoir by a horizontal and a 
vertical face such as R'ft' and Q'P*. 

One of the voussoirs will be projected entirely upon the polygon 
M'N'P'ft'R*, and will, occupy in the right prism having this polygon for a 
base the space comprised betvreen the cylindrical surfaces which form the 
faces of the round tower. The horizontal projection of this and the 
other voussoirs will be obtained by dropping from the various vertices 
of the pentagons, such as M'N'P'Q'R', verticals M'M M", N'N N«, etc, 
of which the parts M M", N N", etc, contained between the two faces of 

-25- 



the eyllndrieal wall will be the horizontal F**oJections of the edges of 
the joints. Since the faces of the wall are vertical cylinders, the 
heads of the voussoirs will be projected horizontally upon the traces 
A e B and C f D. 

28. Patterns ;- Patterns of the joints and of the intradosal sur- 
faces will be required. The latter are parts of a cylindrical surface 
whose right section is A'U'B', and will evidently be shown in their true 
size if that surface be developed. Rectify, therefore, the cuirve of right 
section, by taking upon an indefinite right line X Y (Fig. 70) distances 

b n, n m, etc., equal to the arcs B'N', N'M', etc. , (Pig. 69), which Is 
done by stepping off small chords as many times as necessary. Then, 
having erected perpendiculars to X Y through the points of division, take 
b b' = g B, n n^ = h N, m m* = i M, ... a a* = k A. Through the points 
b', n*, m', .... a* thus obtained pass a cujrve b'n'm'a', which will be 
the development of the curve of the head projected in A e B, which devel- 
opment can be made more precisely by finding intermediate points between 
those above indicated. Similarly, take b b* = g B*, n n* = h N*, m m" = 
I M", ... a a" = k A", and through the points b", n", m", ,,., a" thus 
obtained pass a second curve b'tt'ta'a", which will be the development of 
the curve of the head of the arch upon the inner face of the round tower. 
The true form end size of all the intradosal siirfaces are now perfectly 
known and will be represented by the quadrilaterals b'n*n''b", n'm'm"n", 
etc. 

The joints are plane surfaces and will be obtained by taking upon 
X Y distances n p = N'P', m r = M'R', etc., and erecting through each of 
the points p, r, etc., thus obtained, perpendiculars p p* = 1 P, r r' = 
u R, etc.; p p* = 1 P", r r" = u R", etc. As the intersect ions of the 
joints with the inner and outer cylindrical faces of the round tower are 
elliptical curves, it will be necessary also to determine at least one 
intermediate point between the extreme points tovmd above, which may be 
done by applying the same proced\zre to horizontal right lines (Fig. 69) 
passing, for example, through the middle of the joints N'P', M'R', etc. 

29. Cutting the Toussoirs by the Method of Squaring ; - 

Choose a block of stone having a length at least equal to the great- 
est dimension P v (Pig. 69) of the horizontal projection of the voussoir 
to be cut. The other dimensions of the block should be such that a pat- 
tern of the head M'N'P'ft'R' could be applied upon the two end faces. 
Begin by dressing one of the bases to a plane, and trace upon it the con- 
tour M, ni pi qi rj (Pig. 71) by means of a pattern cut to the vertical 
projection M'N'*P'Q*R' (Pig. 69), taking care that the side M'R' of one 
of the joints coincide as nearly as possible with the quarry bed mi ri 
R2 Mg. 

Then cut away the stone, square with the base already dressed, 
along the line M^ rj, and to the new face apply the pattern m'r'r"m" 
(Pig. 70) of the upper joint, plaeixag it so that m' falls at VL and so 
that r' Is distant from r^ by a length r, Ri = R z (Pig. 69). By means 
of this pattern trace the contour Mj Rj^ R2 % of the joint. Proceed in 
the same manner for the joint which passes through the right line n^^ p^ 
and which shottld be square with the base, and apply to it the correspond- 
ing pattern n'p'p"n" (Pig. 70) in order to trace the contour of the lower 
joint. 

The intrados is a portion of a cylinder passing through the curve 
M ni and whose elements are perpendicular to the base Mj^ n-i pj^ qi r^. 

-26- 



The square will answer for cutting this cylinder, or if its branches are 
too short a templet may be used, cut to the arc M'N' (Fig. 69), and moved 
along the right lines already traced, mi M2. n^^ N2, in such a msuaner as 
to be always parallel to the base M'h'p'q'r . Next apply the pattern of 
the Intrados m*n'n"m* (Pig. 70) to the concave surface of the stone and 
trace the two curves Mj^ N, and Mg Ng. 

The lateral face pj^ q^ Qg ?„ and the upper face Qi Jt*i Ro ^ *"*® 
planes, each of which passes through two known right lines, and are also 
perpendicular to the base of the prism; they can therefore be easily cut 
by aid of the straight-edge. That done, take the distances q^ Qi and qi 
Qg respectively equal to the lengths v P and v P" (Fig, 69) and trace the 
right lines ftx ^1» ^ ^2» *® well as the ouirves Q|| R-i, Qg Rg, using for 
the latter purpose templets cut to the arcs P R, F"R* of^the traces of 
the round tower. 

It remains only to cut the two heads of the troussoir, of which we 
now know the contours U% N^^ P^ Q-, R^ and M^ Ng Pg Q2 ^2* ^ese surfaces, 
being cylindrical, can be cut by moving a Btraight-edge upon the contours 
in such a manner as to pass through points of reference suitably chosen; 
or in such a manner as always to be parallel to the face P^^ Q^ Qg Pg« 

It is possible to cvxt the stone without using the constructions of 
Fig. 70, by noting upon a sketch of the stone the lengths of the differ- 
ent right lines N x, N''x, R z, R"z, P v, P''v, etc., and of some other 
intermediate lines; than, having squared a right prism having for its 
base the contour Mj^ n^ pi q^ r, (Fig. 71), mark upon the stone the dis- 
tances r-j^ R^, rj Rg, nj^ N,, nj Ng, etc., respectively equal to the corre- 
sponding distances given on the sketch, and finally trace by hand the 
curves Mi Rj, R^ Qi, Mj Nj^ etc., for which one or two Intermediate 
points will be needed. 

If the diameter of the arch differ but little from that of the round 
tower, the angles which the vertical faces of the vouasoirs make with 
the inner face of the tower are likely to be too acute, especially to- 
wards the springing lines of the arch, Hiis difficulty may be avoided 
by making the vertical faces perpendicular to the faces of the tower, 
for a few inches in, as indicated for the face F F" (Fig. 69), c d being 
any direction whatever. 

It has here been asstnned that the round tower is of brick masonry, 
an& that no care need be taken to make the Joints of the voussoirs accord 
with the courses of the tower. But if the latter were of stone, the 
joints of the voussoirs might not stop at a circle concentric with that 
of the intrados, but might be given sufficient length to bring about the 
accordance in question. One of the arrangements represented in Fig, 62 
would probably be adopted, but the cutting of the stones would present 
no new difficulty, 

30. A Skew - Arch in a Round Tower with Battered Face and Meeting a 

Spheric aX Dome ;- 
It is assumed that the exterior face of the tower is that of a 
cone of revolution, ot that. In other words, it has a batter, the value 
of which is indicated by the angle Z'C'Y' (Fig. 72), and that the tower 
is covered by a spherical dome. An arched opening is to be constructed 
in the wall, of the tower, having its springing plane on the same level 
with that of the spherical dome. The arch is to be on a skew, that is, 

-27- 



its axis is not to meet that of the tower. 

The vertical plane of projection will be taken perpendicular to the 
axis of the arch, and upon this the semi-circle A'M'B' will represent 
the right section of the intrados of the arch. The horizontal plane will 
be taken in the common springing plane of arch and dome. Let E A B be 
the trace of the cone upon the springing plane, smd P A*B" the face of 
the cylindrical inner face of the tower; it will at the same time be the 
horizontal great circle of the dome. 

Divide the semi-circle A'M'B* into an odd niimber of equal parts, 
five for example, and indicate the joints by right lines R'M'O', P'N'O*, 
etc., all passing through O' in order that they may be normal to the in- 
trados. Limit these joints N'P', M'R', etc., by a circle O'R'D' concen- 
tric with A'M'B*, and terminate each voussoir" by a vertical face P'ft* and 
a horizontal face Q,'R', 

The round tower is assumed to be built of brick masonry; but if it 
were of cut stone it would perhaps be necessary to modify the eon8ti*action 
somewhat and to arrange that the horizontal face of each voussoir should 
be in the same plane with one of the beds of the courses composing the 
tow6r. As it is, each voussoir is vertically projected upon a pentagon 
such as M'N'P'ft'R* and is part of a right prism having one of these pen- 
tagons for a base; further, it occupies in the prism the space between 
the spherical dome, on the one hand, and the exterior face of the round 
tower, on the other hand. It Is necessary, then, to detennine the inter- 
section of this prism with the sphere and with the cone. 

In order to determine the intersection with the cone, imagine through 
the point of which we wish to know the horizontal projection, that for 
example which is vertically projected at M', a horizontal circle, lying 
upon the conical surface. This circle will be projected vertically along 
a right- line parallel to the groiind line and meeting O'Y' at a point m', 
which we project horizontally to m upon G Y drawn parallel to the ground 
line. Then revolve this point into the plane C C" by means of an su*c of 
a circle m m", and through the point m" thus obtained pass a circle m"M 
having the same centre as the round tower. The intersection of this 
circle with the perpendicular dropped from M* will furnish the horizontal 
projection M of this point. 

It will be noticed that the radius of the circle m*M differs from - 
that of the circle E A"B by a distance equal to mx m*. Consequently it 
is not necessary, in order to find the point M, to proceed as above indi- 
cated; but it will be sufficient to draw any right line EPS passing 
through the center of the tower, to take the distance E voq = m'm, , and 
to pass through mg an arc having the same center as the tower, wnich arc 
will evidently be the same as the arc m M previously determined. Similar- 
ly one will find the horizontal projections of the other points where the 
edges of the voussoirs meet the outer surface of the tower, thus deter- 
mining the curve B N M A. The horizontal face Q'R* will cut the cone 
along a cirjeumference having the same center as the round tower; conse- 
quently the arc q R, which Is the prolongation of that which furnished 
the point R, will be the projection of this intersection. The joints 
N'P* and M'R* meet the cone along OTirves P N and R M which pass 
through and on which intermediate points can be found by means which 
have been indicated. Finally, the vertical fac,e P'ft* will cut the cone 
along a curve projected upon the right line P ft. 

in order to construct the head of the voussoir which is upon the 
sphere, dtaw horizontals through M and N' and describe upon the horizontal 

-28- 



plane two circumferences 104 M* and n. N" concentric with A*0"D" and having 
for radii the radius of the latter diminished by the distances m^ g and 
ng h comprised between the vertical D'V' and the arc D*U', which is a 
section of the spherical dome. The same thing will be done for the intra- 
dosal edges of the other voussoirs, thus giving the curve A"M''N'*B" along 
which the intrados of the arch meets the spherical dome. 

As for the joint M'R', the horizontal edge throu^ the point R' will 
pierce the sphere at a point of which the projection R** will be obtained 
by drawing the horizontal R'k r% and subtracting the part k rg from the 
radius of the arc A'CD", describing with the remaiiider an arc which will 
cut the peirpendicular let fall from R' at the desired point R", The 
intersection made by the joint itself will be projected along a curve 
R"M"0", of which an intermediate point can be found by applying the pre- 
ceding method to the middle of the side U'R*. 

An analogous curve P*N" will be found for the lower joint; the 
horizontal face R'^' will give the arc R"S", prolonged from that which 
has served to determine the point R"; and finally the vertical face 
P' ft* will cut the sphere along an arc projected upon the right line P P*. 
But as this last face meets the sphere very obliquely, pass through P" 
a plane P»S" passing also through the axis of the round tower, thus re- 
ducing the projection of the head of the voussoir to M"N*P*S"R'', and giv- 
ing rise to a new vertical face which is shown revolved in the triangle 
a b 0. In this triangle, which is right-angled, the hypothenuse b is 
an arc coincident with A'B^D* prolonged, and the height ,a b is equal to 
that of the vertical face of the voussoir, that is, to P'ft'. 

The vertical face being thus modified it is necessary, in cutting 
the stone, to know its exact shape. Therefore revolve it into the hori- 
zontal plstne, giving the figure p'p"q"q', of which the height p"q" is 
equal to P'ft', and of which the side p'q' is a curve easily constructed 
by erecting perpendiculars from different points of P ft and laying off 
upon these, from p'p", lengths respectively equal to the distances which, 
upon the vertical plane, separate the corresponding points from P', 

31. Making the Patterns :- The development of the cylindrical in- 
trados will be effected by taking upon an indefinite right line distances 
b n, n m, etc., equal to the arcs B'N', N'M', etc.; then, havixig erected 
perpendiculars through the points of division, take b b' = e B, n n' = 
f N, m m* = i M ..., a a* = j A, and the cxirve b'n'm'a* will be the de- 
velopment of the curve projected in B N M A. This can be obtained more 
precisely by procuring interaiediate points between b', n', m', .... a'. 
In the same way take the distances b b" = e B", n n" = f N", m m* = i M", 
...» a a* = j A", and the curve b"n"m''a" will be the development of the 
curve in which the intrados of the arch meets the spherical dome. Then 
the exact fonn of the intradosal surfaces will be known, and will be rep- 
resented by the quadrilaterals b'n*n"b", n*m*m"n", etc. 

The bed joints are plane surfaces, and their patterns will be ob- 
tained by laying off on the development distances n p = N'P*, m r = M'R', 
etc., and erecting through each of the point's p, r,- etc., thus obtained, 
perpendiculars p p'= y P, r r' = z R, etc.; p p" = y P", r r" = z R", 
etc. As the edges of the joints, upon the conical and spherical surfaces, 
are curves, it will be well to obtain at least one intermediate point 
upon each, as by applying the above method to the middle points of the 
joints N'P', M'R', etc. 

-29- 



r 




base 

(Pig. 

line M s. Arrange that one of the joints coincide with the quarry bed. 
Then, upon the face which passes through the aide n^^ pi apply the pattern 
n'p'p"n" (Fig. 73) of the lower joint in such a manner that ni N, = u N, 




Then, through the two known right lines, Pn Qo) Q2 ^* pass a plans, 
upon which trace the contour P^ Qo ^2 ^^ means of the pattern a b e 
(Fig. 72); after which apply upon the cylindrical face cut along m, n-, the 
iutradosal pattern m'n'n*^" (Fig. 73) and trace the curves M^ N^^, JSL N2 
which limit the intrados. Finally, upon the face passing through tne 
right line p, q, mark the contour Pi Qj^ Qg Po by means of the pattern 
p»q'q»p" (Pig. 73), taking care that p. Pi = s P and qj^ Qi = s Q. 

The intrados and all the other lateral faces having been cut, and the 
c-entour^ Mg N„ P_ q^ Rg of the head of the voussolr being then known, 
that head, whTch^ls spherical, can then be cut by employing a templet 
cut to the curvature of D'U' (Fig. 72) applying it always in a direction 
perpendicular to the face ^i Rq ^ % ^1* 

As for the exterior head, of which the contour Mj Ni Pj^ Q, R, is 
also completely known, it is a portion of a conical sx^urface anot can be 
cut by aid of the straight-edge applied in the direction of elements. 
These can be determined by fixing upon the drawing reference points, 
conveniently spaced, which reference points can be obtained by drawing 
from the center of the round tower right lines such as v x (Fig. 72). 

THE DESCENT, OR RAMPANT ARCH. 

33. A descent, or rampant arch, is an arch the elements of whose 
cylindrical intrados are inclined to the horizontal plane. Instead of 
being horizontal as in the ordinary arch. This difference In the posi- 
tion of the elements leads to other differences in the form of the arch 
and sometimes gives rise to considerable dlfficxjlties In the drawing. 

34. A Descent through a Right Wall : - In Fig. 78 let A'M'B* be the 
intersection of the Intrados with a vertical plane A'B* which Is one of 
the faces of the wall. Moreover let B'B" be the base of the ramp, the 
height of which is given. This base and this height form a right-angled 
triangle, .the hypothenuse of which is the direction of the elements of 
the oblique cylinder which ha^ for a base the circle A'M'B*. In order 
to show better this direction take a profile or vertical plane B2 B3 
parallel to B'B", and having revolved it upon the horizontal plane around 
B2 B3 trace there the right-angled triangle B2 B3 C, the hypothenuse of 
which will show the true slope of the ramp. It is clear that this hypoth- 
enuse Bs C will be the trace of the springing plane upon this profile 
plane, and that A''B3 will be the horizontal trace. The two ramps of the 

-30- 



descent are rectangles, projected upon A'E'E''A" and B*P'P"B". 

Divide the face curve into an odd number of equal parts, five for 
example, and through the points of division and the center 0' pass in- 
clined joints and limit these by a circle E'R'P' concentric with the 
first, finally terminating each voussoir by a horizontal face and a ver- 
tical face such as R'Q* and P'Q', Pentagons such as M'N'P'Q'R* will thus 
be obtained which are the upper heads of the voussoirs composing the de- 
scent. The horizontal projections of these same voussoirs are made up 
of the projections of the various parallel edges. In order to determine 
the projections of these edges upon the profile plane, through the points 
M', N', P', etc., draw horizontals M'm, N'n, P'p, etc., and through the 
points m, n, p, etc, where they meet the vertical C Z, pass arcs having 
a common center at C. These arcs will cut the line B2 C D at points 
«ni. ni. Pi. etc., through which draw, parallel to the rampant line C B3, 
right lines mi m2, ni n2, Pi P2» etc., which will be the projections of 
the intradosal and joint edges upon the profile pleme, which projections 
evidently show them in their true length. 

In order to develop the intradosal surfaces for patterns, it will 
be necessary to know the right section of the descent. To this end draw 
through any point Cj of C B3 a perpendicular Ci rs T. This perpendicular 
will cut the edges nj n2, Pi P2. etc., at points ng, P3, etc., which are 
to be revolved upon the vertical ci V by means of arcs ng 04, pg P4, ete. 
Through the points 04, P4, etc., thus obtained, and through the point cj 
draw horizontals, meeting the horizontal projections of the edges of the 
voussoirs at points Mj, Nj, P^, etc. Then trace the curve Aj Mi Bi which 
will be a right section of the descent, and Join by rl^t lines the 
points such as Mt and Ri, Ni and Pi, Ri and Q,, etc., which will give a 
pattern for obtaining the heads of the voussoirs. 

35. Obtaining the Patterns:- In order to rectify the cylinder of 
the intrados, lay off upon the indefinite right line S T (Pig. 79) dis- 
tances S n, n m, etc., equal to the lengths of the corresponding ares 

Bi Ni, Ni Mj, etc., of the right section (Fig. 78); then erect perpendic- 
ulars, upon which take distances S b' = ci 0, n n' = ng ni, mm" = m5 mi, 
etc.; and S b" = ci B3, n n" = ns n2, m m" = mg m2, etc. It will then be 
easy to trace the curves b'n'm'a^ and b''n''m''a", which will be the devel- 
oped face curves of the rampant arch, and which will be identical with 
each other, since the arch is in a right wall. The true size and exact 
form of all the intradosal surfaces will be perfectly known and will be 
represented by quadrilaterals b'n'n^b", n'm*m"n", etc. 

As for the patterns of the Joints, which are plane surfaces, they 
may be obtained by laying off upon the right line S T, from the Intradosal 
edges, distances n p = Ni Pi, m r = Mi Ri, etc., and erecting through 
each of the points p, r, etc., thus obtained, perpendicvilars p p' = P3 pi, 
r r' = rg ri, etc.; p p" = P3 P2» r r" = rs rg, etc. The right lines 
n*p', n"p*, m'r', m"r", etc., will be the exterior edges of the joints, 
that is to say, those which limit them upon the faces of the wall. 
(Certain lines shown in the drawing, but not mentioned, relate to a de- 
scent to be studied later). 

36. Cutting the Voussoirs :- Assiune that it is desired to cut the 
voussoir which has for its head M'N'P'ft'R' (Pig. 78). Begin by squaring 
a right prism having a base mg No P2 12 ^2 (^ig* 80) identical with the 
contour mi Ni Pi Qi Ri (Pig. vB) ana having a length at least equal to 
the distance or"th8 point n2 from the right line u ri. Arrange so that 

-31- 



one of the joints shall coincide with the quarry bed of the stone. Thai 
take upon the prism distances ran M2, P2 P2» ^2 ^2» ^2 ^2' Q<l"al to the 
corresponding distances on the drawing, that is to say, to mz z, pg y, 
r2 8, the last answering for both q2 Q2 *"^^ ^2 %» 

Then, upon the cylindrical face which passes through the arc mz N2 
apply the intradosal pattern m'm"n"n' (Pig. 79) and trace the contour 
M2 N2 Ni Ml . Apply in the same manner upon the faces which pass through 
the right lines m2 r2 and ng P2 patterns m'r'r''m'' and n'p'p"n'' (Pig. 79) 
and trace the contours M2 R2 % % and N2 P2 Pi %• Finally take the dis- 
tances R2 Ri, CI2 ftl> P2 ^1* e<iual to the lengths of the corresponding 
edges measured upon the drawing, edges which are evidently of the same 
length, since, the wall being a right wall, its two faces are parallel. 

The contours of each of the heads being then known, it will be easy 
to cut their surfaces, which are plane. To this end it is only necessary 
to cut away the two truncated prisms which are comprised at each end be- 
tween the base and the adjacent head. The intrados, a cylindrical sur- 
face, will be cut by aid of a straight-edge made to pass through refer- 
ence points suitably chosen upon the face curves, and which can easily 
be obtained frcsn the drawing. 

37. Remarks :- It will be seen from the above directions that pat- 
terns of the intrados and joints are not essential to the cutting of the 
voussoir. 

The joints of the arch, as above drawn, are oblique to the intrados, 
which may be of considerable importance if the slope of the descent is 
great. If it is desired to make them normal to the intrados, proceed as 
follows: Construct the right section Ai M^ B^ (Fig. 78), and after having 
drawn true normals N^ P^, Mj^ R^, terminate them at points Pi, Rj^, where 
they meet horizontals from p^ and r^; then by the reverse or the opera- 
tions hitherto described obtain from Pj,and R^ the points P' and R', 
which will fix the direction of the edges of the joints N'P' and M'R' 
upon the plane of the head. It is tirue that these lines will not be nor- 
mal to the face curves, but this is thought not to be so serious an ob- 
jection as to condemn the plan proposed. 

If the descent were of considerable length and it were no longer 
possible to form each course of a single stone, it would be unwise to 
allow the whole mass of the arch to rest upon two inclined surfaces such 
as the ramps which in Pig. 78 are projected in the rectangles A'A"E"E' 
and B'B'*P"P*, because there woiild be danger of the mass sliding, or at 
least seriously straining by its pressure the arch or other work against 
which It might abut. For this tease, instead of terminating the abutments 
by an inclined plane, prolong the voussoirs of the first course down into 
the abutment, where they are to rest upon horizontal beds, as seen in 
Pig. 81, which is a section along the axis in an arch thus arranged. In 
order to guard still further against sliding, the voussoirs may be given 
a form such as a'b'c'd'e'f 'g'h i ', by which means they are tied together. 
Some simple stones are employed, however, as indicated by diagonals in 
Pigs. 81 and 82, the latter representing a projection of the intrados 
upon a plane parallel to the elements. Such stones are employed for the 
key course, which are the last to be cut and are carefully fitted to the 
space left between the adjacent courses. Figure 83 illustrates the 
general fonn of the voussoirs in this oonstmiction. 

-32- 



The arrangement above outlined prevents sliding, but does not pre- 
vent the intrados and joints from meeting obliquely the planes of the 
two faces. It is therefore an advantage to begin and to end the descent 
with a short horizontal arch, as indicated in Pig. 84. 

38. A Right Descent in a Battered Wall , meeting an Arch in Brick 

Masonry ;- 

The same data will be adopted as for Pig. 78; but as the descent is 
here supposed to meet a large masom'y arch, the springing plane of which 
coincides with the horizontal plane of the drawing, and the first element 
of which is E^P", it will be necessary, in order to define the arch, to 
trace the arc B3 X with a given radius, which will represent a right sec- 
tion of the cylinder made by the vertical plane B2 B3. Further, in order 
to complete the data, trace upon the profile plane the right line G Y, 
directing it relatively to the vertical so as to show the batter of the 
wall. 

Then begin by determining the face curve A* i g B' as follows:- 
Through the point e, where, upon the profile plane, the right line C Y 
meets the edge of the intrados ni n2, drop a perpendicular e f upon C D 
and through the point f pass an arc f h having its center at C. This arc 
meets the vertical G Z at a point h, through which draw a horizontal cut- 
ting at g the vertical from N*. The point g is one point of the face 
curve, and other points may be fotond in the same manner. 

The right section will be obtained exactly as explained in Article 
34, and the intradosal and Joint patterns as in Article 35. The patteims 
will differ, however, a little from those of Pig. 79, since in order to 
obtain them one will no longer lay off frcmi S T the distances comprised 
between ci rg and G D, B3 rg, but instead those comprised between cj rg 
and the right line G Y and the arc B5 X, 

The cutting of the voussoirs offers no new feature, except that the 
head lying in the masonry arch will be cylindrical instead of plane, and 
will be executed as explained in Article 36. 

39. A Skew Descent meeting an Arch in Brick Masonry ;- In Pig. 85 
let A'M'B* be the face curve of the descent, which we suppose situated in 
the vertical plane of the drawing; the right lines A* A", B'B", the hori- 
zontal projections of the springing lines of the intrados of the descent; 
C*D", the first element of the large arch which the descent is to meet, 
and the springing line of which is in the horizontal plane of the draw- 
ing; smd, finally, let C" ms X be the right section of the large arch, 
revolved into the horizontal pliuie. 

First, let there be determined upon the vertical plane which has 
CE for its horizontal trace the projections of the different intradosal 
and joint edges of the voussoirs which form the descent, the joints first 
having been fixed in the customary maziner by dividing the curve A'M'B' 
into an odd ntaaber of equal parts, five for example, and drawing through 
the points of division right lines such as N'P', M'R*, which pass through 
the axis of the descent. 



-33- 



For this purpose, from the point E lay off a length E b2 (Pig. 86) 
equal to the height of the slope, and join bn with C": the right line 
bo C" Till be the comnon projection of the fwc intradosal edges A'A" and 
B B* upon this profile plane. To obtain the projections of the other 
edges of the voussoirs, draw through the points M', N', P', R*, etc., 
horizontals N' nj^ M' m,, P* pj^, R^ r,, etc., and through the points 
oi» »1. Pl» rit 9XC., where these horizontals meet the vertical bo Z, 
pass arcs having for a cannon center the point b2. These arcs will meet 
the line D'C' prolonged at points n2, Bi2» P2» etc., through which draw, 
parallel to b2 C", right lines meeting the right section curve of the 
large arch at points n3,mj.,p2,etc.Fineaiy, through these new points diraw 
horizontals, which cut the horizontal projections of the edges of the 
voussoirs at points N", M», P", etc., which will evidently be the hori- 
zontal projections of the points where these edges meet the large arch. 
Tracing then the curve A*M*N"B* there will be given the horizontal pro- 
jection of the intersection of the intrados of the descent with that of 
the large arch. 

That done, it will be noticed that the faces such as R'ft' must nec- 
essarily meet the large arch along elonents such as R"^". Further, the 
vertical faces such as P'Q' will cut the large arch along curves; but 
these cujrves, since the faces are vertical, will be projected upon right 
lines such as P'Q". The joints will meet the large arch along curves 
P"K'0', R"M*0', etc., which should pass through the point 0' and be tan- 
gent at that point to the right line C*D", 

The vertical projection (Pig. 86) does not give, as in the case of a 
right descent , the true lengths of the edges of the voussoirs, since the 
ground line G'E is not parallel to the horizontal projections of these 
edges. It is necessary, in order to find those true lengths, to procure 
a second vertical projection on a plane parallel to the horizontal pro- 
jecting planes of the edges. It is necessary, moreover, to determine the 
vertical projection of Figure 86 in order to be able to find the hori- 
zontal projections of the faces of the voussoirs upon the large arch, 
which projections will shortly be useful. 

In order to obtain the second vertical projection (Pig. 87), take 
a ground line D'D" parallel to the horizontal projection O'O" of the axis 
of the descent; then, after iiaving marked the height D' Ih of the descent, 
draw the horizontal Di Cx upon which project the points b"'. A', C', to 
Bj, A™, C^, and project upon the ground line the points B", A", 0', to 
B2, A2, C2. Then the rij^t lines D^ D', B^ B2, A^^ A2, C]_ C2 will be the 
projections of the sides of the two ramps; the right line D^ Ci will be 
the vertical projection of the Intersection of the plane which passes 
through the springing elements of the descent with the vertical face of 
that descent; and the ground line D'D"C2 will be the vertical projection 
of the Intersection of the same springing plane with the large arch, which 
intersection is at the same time the springing line of that arch. 

In order to find the vertical projection of one of these edges, that, 
for example, which starts from the point M' (Pig. 85), draw through this 
point the horizontal M' 04, which will meet the vertical P U at m4; then 
through m4 pass an arc described from P as a center and meeting at me, the 
ri^t line P V perpendicular to the right line P Di Ci; then, through 
this new point ms draw a parallel to Di Ci, which will cut at Mi the 
ri^t line M I Ui perpendicular to thi ground line D'D"C2; finally, throu^ 

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the point M draw the right line M. Mo parallel to B, B2f and the point 
Mo, where this parallel meets the peirpendicular M"G I8I2 to the groTjnd 
line D'D" C2, will be the vertical projection of the point where the 
edge in question pierces the intrados of the large arch. Proceed in the 
same manner for the intreuiosal edge which starts from N'; so that the 
curves Bt Nj^ Mj^ C,, B2 No Mo Co will be the lateral projections,- the 
one, of the face cuirve B^N'm'C^, emd the other, of the intersection of 
the intrados of the descent with that of the large arch. The edges of 
the extrados, such as those starting from the points P', Q*, R', will be 
obtained in a similar manner. 

The joint lines upon the plane of the head will be projected along 
right lines Pi Nj, Rj M^, which should pass through the point 0^; and 
those upon the large arch, along curves P2 N2, R2 M2, which ought equally 
to meet at O2. The lines, such as Pg (Jo, proceeding from vertical faces, 
will be curves intennediate points of which can be obtained by means 
already indicated; the other lines, such as Pi Qi, proceeding from verti- 
cal faces, will be right lines perpendicular to the ground line D'D"C2; 
finally, lines such as Ri Qi and R2 ^2 *iH ^^ right lines parallel to 
the ground line D'D"C2. 

In' order to obtain a right section of the descent, draw the right 
line H K (Pig. 87) p erp end i evil ar to the projections of the elements of 
the intrados of the descent; then, after having prolonged (Pig. 85) to a 
convenient length the horizontal projections of the edges of the intra- 
dosal surfaces and of the extrados, draw the right line S T perpendicular 
to those projections; afterward, through the point B^ where S T meets the 
edge B'B" produced pass a line Dg Cg such that the distance u Ag shall be 
equal to c d (Pig. 87), and Bj A5 will be the diameter of the right sec- 
tion. Then make the distances v Mg = c f , x Ng = c e, etc., and through 
the points Aj, M3, N3, Bg draw the curve A3 ... M3 N3 B3 which will be 
the required right section. 

The joint lines will be obtained by making the distances y R3 = c h, 
2 P3 = g, etc., and drawing the right lines M3 R3, N3 P3, etc., which should 
necessarily converge to the point O3. Finally, the upper and lateral 
edges such as R3 Q3 and (^3 Pg will be obtained by drawing parallels 
R3 Q3 to the diameter Ag B3, and drawing other parallels Q3 Pg to the 
axis O'Og of the descent. It will be noticed that the distance ^3 z 
should equal i c. 

40. Patterns of the Development ;- Upon an indefinite right line 
(Pig. 89) take the lengths bs ng, ng mg, etc., equal to the arcs B3 N3, 
N3 M3 of Pig. 88, and erect the perpendiculars bg by and bg bg, ng n? 
and ng no, mg m^ and mg mg, etc., respectively equal to the distances 
c Bj^ and c Bg, e N, and e Ng, f M, and f Mg, etc. The curves b,^ n^ a^ 
and bg ng a,, will be the developments of the two face curves of the de- 
scent, and will determine at the same time the patterns of the intradosal 
surfaces such as m n^ ng mo. 

In order to obtain patterns of the joints, take the distances ng pg, 
mg rg, etc., respectively equal to Ng Pg, M3 R3, etc., (Pig. 88), and 
erect the perpendiculars pg p^ and pg pg, rg r„ and rg ro, etc., equal 
to the lengths g Pj^ and g Pg, h R^ and h Rg, etc. (Pig. 87). The sides 
P7 n.7, r^ my, etc., of these patterns will be right lines, and the sides 
Pg ng, rg mg, etc., will be curves, of which intermediate points can be 
secured By applying the same procedure to the middle point of the sides 
N'P', M'R*, etc., of Pig. 85. 

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41. Cutting the Voussoirs :- Suppose that it is required to cut the 
voussoir which has for a face M'N'P'Q'R' (Fig. 85). Begin by squaring a 
right prism having for a base the contour M3 N, P, Q, R_ (Pig. 88), and 
upon the intrados of this prion apply the patttm m, n„ n^ bio of Fig. 89, 
by means of which trace the contour M^ N^ N^ m^, arranging that the dis- 
tances n^ N-^ and m^ M-^ be equal to j Ni and s Mi (Fig. 87). Simllarly- 
upon the faces which pass through the right lines m r^ and n^ p^ apply tte 
the gat terns m™ rg rg mg and ny p.^ ps ng, and trace the contours M^ r1 

R*^ Wi and N-^ ?•«• pZ N^, making the distances rl r1 and pi pl respectively 
equal to J R, and t Pj^ (Fig. 87). Finally, upon the face which passes 
through % p* apply a pattern cut along Px Qi Q2 ^o (Pig. 87), which is 
the true size of the vertical face of the voussoir; since this face is 
parallel to the plane of projection; and by means of this pattern indi- 
cate the contour pJ- Ql Q'' p^. Then draw the right lines q^ r1, q2 ^2^ 
and the two faces will be completely defined. The first of these, being 
plane, will easily be cut; the second, which is upon the large srch, 
being a portion of a cylinder having its elements parallel to R* Q*. 
will be cut by the aid of a straight-edge kept always parallel to R* Q^, 
or made to pass through reference points conveniently chosen. 

CONICAL VAULTS. 

42. The coniceil vault has its intrados a conical surface. For this 
putTpose of this problem let there be given a right wall, whose faces have 
for horistontal traces the right lines A B and C D (Fig. 91), in which is 
to be constructed sin opening covered by a vault whose intrados is to be 

a right cone having its apex S upon the springing plane, and having for 
directrix the curve A'M'B' traced upon the face A B of the wall, a curve 
which will in this case be t£iken as a seml-circXe. 

It follows from these assumptions that the apex of the cone will be 
vertically projected at the center 0* ot the face curve A'M'B*, and that 
the cone will cut the second face D of the wall along a semi-circle 
C'm'D', of which the diameter C'D' will be equal to the length C D ccan- 
prised between the two springing lines A C S, B D S, 

Begin by dividing the semi-circle A'M'B' into em odd number of 
equal parts, five for example. Then, through the points of division drop 
perpendiculars upon the ground line, which will meet the trace A B of 
the face containing the directrix of the cone at points M, N, etc., which, 
Joined to the apex S, will furnish the horizontal projections Mm, N n, 
etc., of the intradosal edges. The vertical projections M'm', N*n'^ 
etc., of these same edges will be obtained by joining the center 0' to 
the points of division M', N', etc.; then, prolonging these lines by a 
suitable amount, the vertical projections M'R', N'P', etc., of the joints 
will be obtained. Finally, terminate each voussoir by a verticnl face 
Q'R' and by a horizontsil face P'Q*, and the drawing will be complete. 

43. Cutting the Voussoirs :- Take for example the voussoir which is 
vertically projected in the pentagon m'n'P'q'R' (Fig. 91). Begin by 
squaring a right prism having this pentagon for a base, whicli w'll give 

a stone having the form mj nx Pi fti Rx % ^ ^2 '^2 °>2 (^ig* 92). Then, 

-36- 



vipon the front face mark two points M, and N, such that the distances 
Mj^ R, and % Pj^ are respectively equal to the lengths M'R' and N'P' of 
Pig. 91. Through the points Mj, Nj pass an arc traced by means of a 
templet cut to the arc M'N* of Pig. 91, and then draw M, mo and Ni m,. 
The intrados, which is a conical surface, may now be cut by the aid of 
a straight-odge resting continually upon the arc^ Mi % and mo no and 
passing through reference points suitably chosen. The lines u vf x y 
show the direction which the straight-edge should receive in two of its 
positions, which evidently correspond to two elements of the cone to 
which the intrados belongs. The reference points u, v, x, y, are easily 
determined by taking the arcs Mj^ u, u x, x N, , m2 v, v y, y n^,, respect- 
ively equal to the arcs M'u, u x, x N', m'v, v y, y n' (Pig. 91), 

44, Vertical Conical Vault : - This name is given to a vault the 
intrados of which is a right circular cone with vertical axis. The con- 
stnaction may be used to cover a cylindrical tower in order to form a 
spire on the summit. In Pig. 93 let AND and B M C be the horizontal 
traces of the two faces of the tower. Begin by fixing the apexes S' and 
8* of extrados and intrados, taking care so to place them, relatively to 
each other, that the thickness of the conical vault diminish as the apex 
is approached. This is in order to lighten the weigtit of the vault with- 
out lessening its stability. It is evident that the apexes S* and s' 
should be situated upon the axis of the tower. Then proceed with the 
arrangement of the vault, employing for bed joints conical surfaces per- 
pendicular to the intrados and with their respective apexes, situated upon 
the axis of the cone. Porm the vertical joints by planes passing through 
the axis of the cone, and arrange them to be continuous in alternate 
courses. In order to avoid the acute smgle F'E'B' the construction 
K'g'H' may be adopted. 

45. Putting the Voussoirs :- First Method : Prepare a stone with six 
faces, two horizontal and four vertical. The two horizontal faces r y x u 
and 8 t V (Pig. 94) will have the form of the pattern h h''i"i (Fig. 93) 

on which is horizontally projected the voussoir in question, and will be 
distant from one another by an amount equal to m"m'. Tire vertical faces 
will be composed of two concentric cylinders u t v x, y r s, and of two 
rectangular planes ruts, y x v. The stone will thus take the form of 
a stone in a vertical cylindrical wall. 

Now apply upon the two vertical plane faces a pattern cut to the 
contour a'b'd'c', taking care that the four vertices are conveniently 
placed, that is so that b s = b'm', c t = c'n', d u = d'n", and a t = 
a'n*. Then by means of templets trace the arcs d di, a a^, and with a 
flexible straight-odge applied successively upon each cylindrical face 
trace c ex and b bj (the latter is invisible upon the figure). Finally 
divide into the same number of equal parts the four arcs a a^, c Ci, 
d d^, b b^, which will funiish points through which to pass a strai^it- 
edge in executing the four conical surfaces between which the stone is 
ccanprised. 

The above method of cutting entails considerable loss of stone, to 
avoid which the following method has been applied: 



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46. Second Method :- In Pig. 95 let B D D"B» be the horizontal 
projection of one of the voussoirs composing a right conical vault. Sup- 
pose this voussoir cut, midway of its length, by a plane parallel to 
the vertical plane and containing the axis of the vaiilt, so that the two 
parts of the stone as thus cut are vertically projected along the same 
lines. In the figure that portion in front of the secant plane is 
(Knitted in elevation, for the sake of clearness. 

It is seen that the vnussdir is comprised within a right prism hav- 
ing for a base the quadrilaterea h'D'e'A* and the length of which is 
equal to the distance which separates, upon the horizontal plane, the 
two parallels v x and v"x", drawn through the points b and b*, procured 
as below. 

The horizontal projections of the curves in which the surfaces of 
intrados and extrados are out by the two planes o e', oi A', perpendic- 
ular to the vertical plane, are the curves b A b", h G h", P e P", 
D c D", which determine the points b and b" through which the two par- 
allels V X and v*x" are passed. 

If now we revolve the two planes o e* and o^^ A' about x x" and 

V v" as axes, vintil they are pars^lel to the hortzontfO. plane, we shall 
obtain in true size the two contours D D'fi^fj^ and bj bi*hj_*hi, along 
which these two planes meet the different faces of the voussoir, pro- 
longed. 

Next describe from S' as a center, with radii S'E', S'A', two con- 
centric arcs, and making them respectively equal to the horizontal arcs 
BAB" and PEP" there is obtained the pattern P P'R'R, which will be 
the development of the conical face of the extrados. Proceed in a 
similar manner to obtain the pattern M M'N'N, the development of the 
conical face of the intrados, taking s' for a center. 

These constructions having been cc»npleted, begin by squaring a 
right prism (Pig. 96) of which the base a2 h2 d2 e2 is equal to the con- 
tour A'h'D'e' of Pig. 95, and of which the length e2 63 is equal to 

V v"; then, upon the upper and lower faces trace the contours r s f 3 f 2 
and m n b3 b2 by means of the patterns bj. bj^hi "h^^, and D D"f]^"fj^ (Pig. 
95), taking care that d2 r = x D and h2 m = z n. Next cut the two plane 
faces destined to foxm the vertical Joints, being guided for each by 
the four right lines which limit it; after which dress the conical sur- 
faces of intrados and extrados, being aided by a straight-edge, which 
always rests upon the cui^es limiting the faces, at reference points 
suitably chosen, as indicated upon the drawing. When these two surfaces 
have been cut, apply to them the two patterns of the development M M'N'N, 
P P'R'R, making them take the curvatui'e of the surfaces, and trace the 
curves which are to seirve as directrices of the second conical surfaces 
destined to torm the bed Joints, as shown in Pig. 97. 



THE SPHERICAL DOME. 

47, A spherical dome or vault is one whose intrados may be con- 
ceived as formed by a quarter-circle turning about one of its limiting 
radii, assumed to be in a vertical position. The simplest and the usual 
mode of arrangement of the voussoirs is in courses cranprised, as regards 

-38- 



p 



the Intrados, between horizontal planes. Consequently the lines which, 
upon the intrados, separate the courses, are horizontal circles, projected 
horizontally in circles of the same size as the originals, and projected 
vertically in right lines parallel to the ground line. The lines which 
divide the courses into voussoirs are circular arcs resvilting from the 
intersection of the vault with vertical planes passing through the center 
of the hemisphere. These lines are projected horizontally in right lines, 
and vertically in elliptical ares. 

In Pig. 98 let A'C'B* be the semi-circle resulting from the inter- 
section of the intrados of the dome with a vertical plane parallel to 
the vertical plane of projection; let A D B be the horizontal projection 
of the circle forming the springing line of the done (in that half of 
the horizontal projection which is above the diameter a b the spectator 
is assumed to be under the dome, while in the other half he is supposed 
to be above the dome, and the lines which are full in one half of the 
plan Are thus broken in the other half) ; further, let a R b be the circle 
of the base of the vertical cylinder which forms the exterior face of 
the cylindrical wall upon which the dome is constructed. 

In order to arrange the extrados of the dome describe a circular 
arc k'f'p' having its center upon the vertical f'O' and below the point 
0', so that the thickness at the key C'f shall be consistent with the 
dimensions of the dcme, the thickness increasing toward the springing 
line. 

Divide the curve A'C'B' into such an odd number of equal parts as 
will give the desired nianber of courses. Then, through the points of 
division imagine horizontal planes to be passed cutting the dcnne in cir- 
cles, which are horizontally projected in circles and vertically project- 
ed in parallels to the ground line, as shown in the figure. This system 
of lines will form the intradosal edges of the bed joints. In order to 
divide the courses into voussoirs, cut the dome by a series of vertical 
planes passing through the axis O'C' and arranged equidistant for the 
sake of symmetry. These planes, A 0, E 0, 6 0, etc., will cut the sphere 
along great circles, which will be vertically projected in ellipses, such 
as G H'I'C'. This curve is constructed by projecting upon the vertical 
plane the points S, H, I, where the trace G of the corresponding secant 
plane meets the different circles which form the intradosal edges of the 
bed joints. These ellipses should be broken upon alternate courses, as 
indicated upon the drawing, in order more firmly to bind the voussoirs 
together. 

The vertical joints as above determined will be nonnal to the sur- 
face of the intrados, and in order that the bed joints shall also be 
normal to that surface let them be formed by cones having their apexes 
at the center of the sphere, and having for their directrices the cir- 
cles which have been adopted as the intradosal edges of the joints. 
These cones will cut the surface of the extrados, which is also spheri- 
cal, along horizontal circles, which will be horizontally projected in 
circles equal to themselves and vertically projected in parallels to the 
ground line. Thus the cone which has for its directrix the circle K HP, 
K'P', will cut the surface of the extrados along a circle projected 
horizontally in k h p and vertically in k'p*. 

If we consider a particular voussoir, that for example in the second 
course between the meridian planes A and G, we see that its horizontal 

-39- 



projection is in the contour k N I h and its vertical projection in the 
contour k'K'H'I'i'n' . The last course should be composed of a single 
stone which is to form the key, although this can be omitted, if it is 
desired to leave an opening, without compromising the stability of the 
dome. 

48. Cutting the Vouasoirs by the Method of Squaring : - Begin by 
squaring a right prism u y x y u^'x'y' (Fig. 99) which has its two 
bases u v x y and u'v'x'y' equal to the pattern N I h k of the horizontal 
projection of the voussoir which it is proposed to out, and of which the 
height X x' is equal to n'e'. Then, upon the lateral faces u u'y'y and 

v v'x'x of this prism trace the contours Ki N^ ni k^ and H^ Ii Ij hi, 
employing for this purpose a pattern cut to the contour K'N'n'k' of Pig. 
98 (The same pattern will answer for slLI the voussoirs of a single course, 
which voussoirs can indeed be considered as generated by the rotation of 
the pattern about the axis of the dome). In order properly to apply this 
pattern, care should have been taken previously to mark upon each of the 
faces two reference points N^ and n]_, Ii and ij, by making N^ u = Ii v = 
I'm', and n^ u' = i^ v* = N n. 

Next, upon the front cylindrical face, trace the circular arc Nj Ii 
by means of a flexible straight-edge applied upon the cylindrical surface; 
similarly upon the rear cylindrical face, with the same straight-edge, 
trace the circular arc kj hi . Finally, upon the lower and upper faces 
trace the two arcs K^ H^ and nj i^ identical with the arcs K H and n i, 
using for this purpose templets cut along these latter arcs. 

The tracing being thus completed, begin by cutting the spherical 
intrados by moving upon the arcs Ni Ii and Ki H^ a templet cut to the 
curvature A'K'N'B*. Care is to be taken that in all its positions 
this templet correspond to a meridian plane, which can be assured if 
the arcs Nx I^ and KiHjbe first divided into the same number of equal 
parts. The upper and loirer joints,, which are conical surfaces, will be 
cut by employing a straight-edge, to be moved upon the two arcs which 
limit each svirface, these arcs having first been divided into the same 
number of equal parts. The extrados will be cut by aid of a concave 
templet cut along the arc k'n', but most often this surface will be left 
more or less rough. 

The method which has been outlined is believed to be the most exact 
in its results, but it has the serious disadvantage of wasting a great 
deal of stone and a great deal of labor. 

49. Second Method by Squaring ;- This method is analogous to that 
previously described for the vertical conical vault. First project the 
voussoir which it is proposed to cut upon a meridian plane dividing it 
into two symnetrical parts, as shown in Pig. 100. Then cut a right 
prism having for its base a pattern of the vertical projection M'N'P' 
p'q'm' and a length equal to n ni, after which lay off on the edges of 
this prism the lengths S m, U M, V N, and V Nj^, X q, Y p, and Y p^, Z P 
and Z Pj, and then mark upon the two bases the points n, n', and ni, n^', 
thus furnishing four points for each of the vertical planes n P and n^ P^, 
that is to say, four right lines which shall serve as guides in cutting 
the two plane faces. When these have been cut, trace the contour of the 
vertical Joints by means of a pattern cut along the principal meridian 
section M'Q'q'm'. Finally, upon the horizontal faces q'p' and M'N' 
indicate the circular arcs p q Pi and N M N^. 

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One may then out the upper oonioal joint, of which are known the 
two extreme elements P p, P'p' and P^ p^, P'p* and a directrix p q p, , 
p'q', by employing a straight-edge, made to glide upon the directrix^ 
and always converging with the two extreme edges. The work should be 
verified by applying a flexible pattern cut to the development of the 
conical Joint. When this pa-ttern has been applied to the concave sur- 
face it will remain only to trace upon the surface the arc P Q Pi, P'(i', 
following the lower edge of the pattern. The lower conical joint will 
be executed in the same manner, using as directrix the arc n m ni, m'n', 
already traced, and the corresponding pattern from the development, 
which pattern will permit of indicating the circular arc N M N, , M'N'. 

It then remains only to cut the spherical intrados, employing for 
this purpose a templet cut to the curvature of the principal section of 
the dome, and made to glide upon the two parallel arcs P ft P^, Q'P', 
smd N M Nj, M'N*. The templet should in each position pass through ref- 
erence points fixed in advance by dividing each of the two arcs into the 
same number of equal parts. 

This method of cutting, though avoiding certain disadvantages of 
the first, is less exact. 

50. The " Bowl " Method ;- If we join (Pig. 98) the point K to the 
point H and the point N to the point I, we shall obtain two right lines 
K H and N I which are evidently parallel and which are consequently in 
the same plane; further, they sure the two bases of a trapezoid K H I N 
of which K N and H I are the other two sides, and of which K I is one 
of the diagonals. But this trapezoid is the projection of another trap- 
ezoid the four vertices of which are upon the intrados of the spherical 
dome . 

It is necessary first to determine the true length of each of the 
sides of the trapezoid and of the diagonal which meets the vertices K, 
K', and I, I'. Now the side which unites the vertex N, N', to the vertex 
I, I', being horizontal, is evidently equal to Its horizontal projection 
N I; and the same is time of the opposite side, which is equal to its 
horizontal projection K H. The other two sides are equal, and have for 
true length the distance of the point K* from the point N , that is to 
say, the chord of the arc K'N*. Finally, the diagonal has for its true 
length the hypothenuse I y of a right-angled triangle I y K, constructed 
upon I K as a base and with a height y K equal to m'l', that is to sdy, 
to the difference of height of the points K, K', and I, I', above the 
horizontal plane. 

This granted, after having leveled one of the faces of the block of 
stone which has been chosen, construct upon it a trapezoid identical 
with that which has been mentioned, in the following manner: Trace, in 
the first place, a right line a b (Pig. 101) equal In length to K H; 
then, from a as a center, with a radius equal to I y, describe the arc 
u v; next, from b as a center, with a radius equal to the chord of the 
arc K'N', describe the second arc x z, which will meet the first at a 
point o; through this point c draw a parallel c d to a b, upon which 
take a length c d equal to N I; finally, join the point d to the point a. 
The trapezoid thus having been constructed, pass a circle through Its 
fovir vertices a, b, c, d, of which the center o will be at the intersec- 
tion of perpendiculars r o and s o erected at the middle points r and s 
of the sides d e emd da. 

-41- 



Assume this eonsti*uetion to have been completed upon' one of the 
faces of a stone, as shown in Fig. 102. Then out out the stone within the 
circumference N^ I]. K^ K-,, giving the concavity the form of a sort of 
bowl, having the same radius as the spherical dome. To do this, employ 
a templet cut to the curvature of a principal section of the dome, taking 
care that the templet rest always upon the circvanference Nj In Hi Ki ; 
that its plEuie be always maintained perpendicular to the plane or the 
circumference; and that in each of its positions it pass throvigh two 
reference points, marked in advance, at the extremities of the same 
diameter, such as the points e and f, g and h. 

When this bowl has been completely and accurately cut, trace within 
it the tirue contour of the intrados of the voussoir. To this end cut a 
templet to the curvature of the principal section of the d<xne, and 
chamfer the edge, if thick, for greater precision; apply it within the 
bowl in such a manner that it coincide perfectly with it, and that it 
rest exactly upon the points Ii and Hj^; and trace the are Ij H^. Simi- 
larly, with the same templet trace the arc Nj^ K,. Finally, with other 
templets cut upon the parallels N I, K H, trace the arcs N^ I^, K-^ H^. 

The contour of the intrados being then perfectly indicated, cut the 
bed Joints and vertical joints. For this purpose use a bevel with one 
branch curved and the other straight. The curved branch a b (Fig. 103) 
will be cut along a great circle of the sphere which forms the intrados 
of the dome, that is to say, tilong the cui^e A'K'C'B'. The straight 
branch will be fixed to the curved one In such a manner that its edge 
b c is peirpendicular to a tangent to the curve a b at b. In using this 
bevel apply the curved branch in the cut bowl, taking care that the plane 
formed by the two branches be always perpendic\;ilar to the contour of the 
intrados. 

Then cut away the stone along the four arcs which form the contour 
of the intrados, and form thus the four joint faces, which should co- 
incide with the straight branch of the bevel, applied as above directed. 
The extrados, as previously stated, is usually left rough. 

51. Method by Intradosal Patterns ;- This method' is applicable oialy 
when the radius of the dome is considerable, say 20 ft.; it is then very 
advantageous, being much simpler than the preceding method, and present- 
ing more precision than the employment of templets for marking the out- 
line of the intrados, which always leave some uncertainty. It is fovmded 
upon the principle that, in the case of a dome of large radius, the 
meridian arcs, such as K'N', which determine the height of the intrados 
of each course, differ but little from their chords; so that one can 
consider, without appreciable error, the intrados of each course as being 
a portion of a cone of revolution, which has its apex upon the axis of 
the dcme, and which would be generated by the chord of the meridian arc 
corresponding to the course in question. 

Now the cone is developable; consequently, if we prolong the chord 
P'S' to the point V, where it meets the axis of the spherical dtane; 
if, further, from this point V* as a center, with V'P' and V'S* for radii, 
we describe the circular arcs P*z and S'x; and if, finally, upon these 
arcs we take lengths respectively equal to P v and S u, we shall obtain 
a pattern P'z x S' which will be the development of the* intrados pro- 
jected horizontally In P S u v. 

-4S- 



The pattern of the intrados having been constructed as directed, 
apply it in the bovl (Fig. 102) cut as has been previously explained, 
making the pattern coincide vith the concave surface, vith its four ver- 
tices at the points Nj, Ij, H^, Ki, marked in advance upon the circle 
e h f g which limits the bowl. At a single operation the contour of 
the intrados can then be traced, after which complete the cutting of the 
voussoir as directed under the preceding method. 

GROINED AND CLOISTERED VAULTS. 

52. The Groined Vault : - This term is applied to a vault the in- 
trados of which is formed by the xinion of the soffits of two arches which 
meet, these arches having the same springing plane and the same rise 
(Fig. 104), The cylinders of the soffits of the two arches intersect 
along two curves or groins , which are pleme curves, and which conse- 
quently are projected horizontally in right lines which are the diagonals 
of a parallelogram formed by the inner faces of the abutments, assuming 
the two arches to be circular or elliptical. It will be seen from the 
above, that the right sections of the two arches which form a groined 
arch are dependent, one upon the other, and that, one being given, the 
other must be consistent with it. 

Let the two arches be as shown in Fig. 104. Upon A'B* as a diameter 
describe a semi-circle A'L'B', which will be the principal section of 
one of the arches, and draw the diagonals AG, B D, upon which will be 
P'-o.Jected the curves of the intersection of the two arches. Now divide 
•...*e semi-circle described upon A'B' into a suitable odd number of equal 
parts, and through the points of division pass planes M'O'O, N'O'O, etc., 
passing through the axis O'O of the arch. These planes, which furnish 
the joints, will cut the intrados along right lines which will be the 
intradosal edges, and which will be horizontally projected in parallels 
to the axis 0; but the continuity of these parallels will be inter- 
rupted in the angles A D and BOG. Through the points such as m and n 
where these projections meet the diagonals draw, parallel to the axis 
0" of the second arch, right lines such as m Mi and n Ni , which will be 
the horizontal projections of the edges of the joints of this second arch. 
It will be noticed that these right lines also undergo a break of con- 
tinuity in the angles A B and COD. 

Take then a ground line Aj' Dj', perpendicular to the axis 0", 
above which lay off the ordinates u-^ Mi', v, Nj', x^ L^*, etc., respect- 
ively equal to the ordinates u M', v N*, x Li', etc., of the principal 
section A'M'N'B'; and through the points A^', Mj^', Nj', L^', etc., pass 
a curve Ai' M-i' N, ' Ln' Di', which will be the principal section of the 
second arch and will here be a semi-ellipse. Then, through the same 
points, Ml', Nt', Li', etc., draw right lines normal to the curve Ai' 
M,' Ni' D^ , Which will be the vertical traces of the joints of the 
second arch. 

In order to constiract these normals one may employ a property of 
tangent planes. For this purpose, at the point M' draw the tangent F'M* 
to the semi-circle A'M'B'; through the point F* where, this tangent meets 
A'B' prolonged draw a parallel F'f to the axis O'O, which will meet at 



-43- 



f the diagonal C A prolonged; through the point f pass another right 
line f Fi ', parallel to the other axis 0", which will cut at Pi ' the 
ground line Di' Aj'; finally unite the two points Fi* Mi' by a right line, 
which will be the tangent at Mj ' to the curve of the second arch. It 
will only remain then to draw through the point Mi ' a right line Mi's 
perpendicular to this tangent and one will have the normal at the point 
Mj^'. Proceed similarly for the other normals. 

The vertical joints will be formed by planes perpendicular to the 
elements of each arch, care being taken to make them alternate in the 
courses, as indicated in the drawing. Tlien the polygon R N n Ni R^ z 
will be the horizontal projection of one of the groin stones, that is to 
say, of one of the voussoirs contiguous to the groin curve A C and 
consequently forming at the same time a part of both arches. These are 
the only voussoirs that require special mention, since the others, not 
participating at the same time in both arches, will be cut like those of 
any simple arch. 

To form the extrados of the vaxilt, proceed as follows: 
After having determined as usual the rl^t section G'R'ft'P'H'I' of the 
extrados of the first arch, construct frraa it the right section of the 
extrados of the second arch Just as was done for the intrados, that is 
to say, for each point such as Pi ' take an ordinate y;|_ P]^' eqtial to the 
corresponding ordinate y P' of trie first arch. 

In order to determine the horizontal projections of the edges along 
which the bed- joint planes meet the extrados, let fall from the extremi- 
ties of the joints perpendiculars Pj^' yi and P'y upon Aj' Dj^* and upon 
A'B', and draw through the points yi and y right lines yx p and y p, 
respectively parallel to 0*0 and O'O. These right lines will be the pro- 
jections in question and will meet at a point p, which Joined to the 
point n will fvirnish a right line p n, which will be the projection of 
the right line along which the joints N*P' and Ni' p,* intersect. 

If it is desired to know the true shape of the two Identical groin 
curves A C and BOD, tlrirough t^e points U, V, X, 7, etc., erect per- 
pendiculars U Ug, V V2, X X2, etc., respectively equal to u M*, v N', 
X L', etc., and through the points U2, V2, X2, pass a curve B U2 V2 D, 
which will be the curve camion to the two groins. 

53. Gutting the Voussoirs ;- Begin by squaring a right prism of 
which the base a b c d e f (Fig. 105) is identical with the horizontal 
projection R N n Nj^ R, z of the voussolr which it is desired to cut, and 
of which the height Is equal to K Pi'. Upon the front face mark the 
contoiir S''T''M"N"P"(l"R" of the head of the voussolr by means of a pattern 
cut to the contour S'T'M'N'P'ft'R* (Fig. 104). Similarly, upon the lat- 
eral face e e'd'd trace the contour S"'T"'M"*N"*P"'q'"R"' by means of a 
pattern cut upon Si' Ti' M, ' N,' Pi* fti' Ri', after which draw the hori- 
zontals M-ta" and m*M«"7 S»2' and z*S"', R»z and z R"', N"n" and n«N'',etc. 

Then out the cylindrical intrados which passes through the arc M"N" 
by using a templet cut to that arc, which is to be moved along the -fewo 
right lines M'ta" and N"n", making It pass through reference points marked 
upon each of those lines. Cut in the same manner the Intrados of the 
elliptical arch, which passes through the arc M"* N"' and meets the 
intrados previously cut along a curve m'n" which will be a portion of 
the groin. The other faces, which are plane surfaces, will be easily cut. 



-44-' 



It has here been assumed that the arches meet at right angles; but 
this meeting may be at any angle whatever, as seen in Pig. 106, where the 
cylinders of the intrados alone are represented, without the mode of 
procedxire being thereby changed. Only, the two groin curves, instead of 
being identicaa, will be unlike; and the edges of the intrados, instead 
of being perpendicular to each other, will make an angle equal to that at 
which the two arches themselves intersect. 

54. The Vault with Double Groin ;- illustrated in Pigs. 107 and 
108, but which will not be described in detail, is a vault whose groins 
are intersected by horizontal cylinders, springing trcm the corners of 
the abutments and united at the top of the vault by a flat vault having 
the foiTn of a parallelogram. 

55. The Cloistered Va\ilt ;- This, like the groined vault, results 
from the intersection of two cylinders, having their axes in the same 
horizontal springing plane and having the same rise; only it differs in 
this, that the parts of the elements which are preserved in one are pre- 
cisely those which are suppressed in the other. 

Thus (Pig. 109) A B C D being a rectangular space which is to be 
vaulted, one adopts for the intrados a surface formed of four parts, 
belonging two and two to the ssune cylinder; the parts projected horizon- 
tally upon the triangles A D and B C will form part of the same 
cylinder having for right section the semi-circle A G'B'; the other two 
parts, projected upon A B. and COD, will belong to another cylinder, 
of which the right section will be the curve Bi ' Gi' Cj', the point Gi' 
being at the same height above the springing plane as the point G*. It 
resTilts from this new combination that the two groins projected upon 
A C and B D, instead of being salient, as in the groined arch, are 
re-entrant. 

The general mode of arranging the voussoirs in the cloistered vault 
does not differ from that in the groined vault. The bed joints are alike 
furnished by planes passing through elements of the cylinders, and the 
vertical joints by planes perpendicular to those elements. 

The cutting of the voussoirs at the angle, the only ones which 
present special difficulty, is effected in the same manner for both 
vaults. As in the groined vault, so in the cloistered, only one of the 
two arch sections can be arbitrarily chosen, and the other must conform 
to it. The key, and even one or more of the adjacent courses, can be 
(fitted, if desired, without compromising the stability of the vault. 

In architecture it is sometimes foiind desirable to flatten the 
upper part of the va\ilt, as illustrated in Pig. 110. 



-45- 



LUNETTES. 

56. Lijnettes are vaults formed by the penetration of one arch Into 
another, or indeed Into any vault whatever, having the same springing 
plane but a different rise. In this respect they resemble constructions 
which have already been studied in which one arch penetrates another. 
They differ in that, in the Lunette, the courses of the penetrating arch 
must Join those of the arch which it enters, which is supposed to be in 
this case, not of brick masonry, but of cut stone. 

57. Right Lxinette in an Arch: - This l\inette is formed by the 
meeting of two arches whichTntersect at a right angle, having the same 
springing plane, but a different rise. In Pig. Ill let A'M'B* be a right 

.section of the intrados of the smaller arch, and A''M"N" of the large 
arch, . the latter section being here revolved into the vertical plane. 
The axis of the small arch is the right line O'O; that of the large arch 
is the right line o O^. 

Begin by dividing the principal section of the large arch into an 
odd number of eqxxal parts, and proceed similarly for the small arch, 
teUcing care, however, that the first point of division L* be situated a 
little lower than the corresponding point L", for reasons which will 
shortly be explained. Then determine, as ordinarily, for both arches the 
bed joints and the extradosal surfaces. 

The horizontal projection of the line of intersection of the two 
arches will be determined by cutting the arches by horizontal planes. 
Thus, if it is desired to find the point M of this projection, drop from 
M* upon the ground line the perpendicular M'M; similarly, from the corre- 
sponding point m" of the large arch drop upon the ground line a perpen- 
dicular, revolving the foot to mi by means of a circular arc; then 
through the point mi draw a horizontal mi M, which will meet the first 
perpendicular M'M at a point M, which will evidently be a point of the 
projection in question. Thus one will obtain as many points of the pro- 
jection as are thought necessary. 

When this projection ALMS has been determined, find the lines 
along which the joints of the small arch cut the intrados of the large 
arch. Now if the joint M'q', in particular, be considered it will be 
seen that it cuts the intrados of the large arch along a curve M m, which 
will stop at the point m, where it meets the intradosal edge m S, which 
corresponds to the point M" of the right section of the large arch.- 
This cui^e will be a portion of an ellipse, which, if it be prolonged 
beyond the point M, ought to pass through the point 0, which will be 
its sunanit, and consequently to be tangent at that point to the spring- 
ing line A B. 

From the point m, the joint M'Q' cuts the joint M'q* of the large 
arch along a right line m q, of which the extremity q is obtained, as 
shown upon the drawing, by means of a horizontal plane Q'q' passing 
through the point Q". It will be noticed that this right line m q ought 
to pass through the point o, where the axes of the two arches meet, 
since here these axes are the traces of the two joint planes, the arches 
being supposed full -centered, 

-46- 



Through the point q draw, parallel to the axis of the large arch, 
a right line q T, which will be the extradosal edge of the joint M*4" 
of the large arch, which joint will have for its horizontal projection 
m q T V. On the other hand, the Joint M'Q* of the small arch will cut 
the extrados of the large arch along a curve q Q, each point of which 
will be obtained by means of horizontal secant planes, as indicated on 
the drawing, and this joint M'Q' will have for horizontal projection the 
contour M m q Q e f . The curve Q P z will be that along which the extra- 
dosal surfaces of the two arches will meet. 

It is easy to see why the first point of division L' of the rigbt 
section of the small arch ought to be lower than the corresponding point 
L" of the large arch. For if It were othen^ise, there would be Joints 
of the small arch which would cut the intrados of the large arch, and 
vice versa, uselessly complicating the cutting of the voussoirs, and 
producing lines which would be disagreeable to the eye. 

It will be noticed that the line projected in A M B (Fig. Ill) is 
only a kind of groin curve, and the cutting of the voussoirs is so 
closely analogous, to the corresponding operations for the groined arch 
as not to require special explanation. In Fig. 112 is given a view of 
the voussoir which has for its face, in the small arch, the contour 
M'Q'P'R'; and in Fig. 113 is given a view of the voussoir forming the 
key. 

It frequently happens that the lunette alone is of cut stone, and 
that the two arches are of brick masonry, except the voussoirs at the 
groins, as seen in Fig. 114. The drawing is constructed in a manner 
similar to that already explained, but pare is to be taken that the 
joints of the stone voussoirs coincide upon the arches with one of the 
bed joints of the courses of bricks which compose the masonry of those 
arches. 

58. Skew Lunette in an Arch : - TOiat has been said regarding the 
right lunette applies also to the skew lunette. The drawing, and the 
cutting of the groin stones, are executed in the same general manner. 
The only difference between the two oases is that the angle between the 
axes of the arches is right in one case and acute in the other. There 
is an objection to the skew lunette in that the angles which the intra- 
dosal surfaces of the voussoirs make with each other are aeute, at least 
for the voussoirs situated on the side where the abutments of the two 
Arches meet at an acute angle. To avoid this difficulty, the small 
arch A (Fig. 115) may be stopped at a vertical plane m n, and at this 
plane an elbow constructed so as to give the arch a direction perpendic- 
ular to that of the large arch B, thus replacing the skew lunette by a 
right lunette. 

59, Skew Lunette in a Spherical Dome : - In Fig. 116 let A 02 B 
be the springing circle of the spherical dome, and aj A and bj B the 
two springing lines of the arch which penetrates the spherical dome, all 
these springing lines being situated in the same plane, which will be 
adopted as the horizontal plane of projection. The vertical plane of 
projection will be taken as a plane passing through the center of the 
spherical dwne, and parallel to the eixis o^ 02 °^ t**® arch, 

-47- 



This vertical plane will cut the spherical d<»ne along a meridian 
isection A"M''N", which will be divided into equal parts A"!", L^M", InfN", 
etc. Upon this same vertical plane revolve the right section of the 
arch, along A'L'M'N*, and divide it into equal parts A'L', L'K', M'N', 
etc., BO that the first point of division L' is lower than the corre- 
sponding point L" upon the spherical dome, for reasons explained in 
Art. 57. Finally, arrange the extradosal surfaces of the two vaults as 
ordinarily, and indicate the joints of each of them. The Ivmette is 
sk^ff, since the axis o^ 02 of the arch does not pass through the center 
of the spherical dome. 

Now determine the horizontal pi^ojection of the curve along which 
the arch cuts the dome. To this end draw horizontal secant planes, 
such as M'm", This latter will cut the intrados of the arch along two 
intradosal edges, one of which, n^ M, will meet at M the circle along 
which the intrados of the spherical done is cut by the same horizontal 
plane M'm", This point M will be one point of the intersection in ques- 
tion, and in a similar manner can be obtained as many other points as 
thought necessary, 

/When the curve A M N B, has been determined, proceed to find the 
lines along which the joints of the arch cut the intrados of the spher- 
ical dome. Now, if the joint N'P' be considered in pai^tieular, it will 
be seen that it cuts that intrados along a circular arc, projected in 
the elliptical arc N Q, The point Q is obtained, as above, by means of 
a horizontal secant plane Q'N". If the curve be required with greater 
precision, intermediate points can be obtained by the same constniction. 
Further, it will be seen that the curve should pass through the point 
02, and should be tangent at this point to the springing circle of the 
spherical dome. 

The same joint N'P' cuts the conical joint of the spherical dome 
which is foimed by the rotation of the normal N" about the vertical 
axis of the dome, along a curve Q S. In order to obtain points of this 
curve, such as S for example, employ a horizontal plane S P", which will 
cut the joint of the arch along the right line s, S, and the conical 
joint along a circle f S having its center at 0; that right line and this 
circle will meet at & point S which will be the point in question. 
Other points can be obtained in the same manner. 

After having cut the conical joint, the joint N'P' cuts the extra- 
dos of the spherical dome along a curve S R, the different points of 
which will be obtained by means of horizontal planes. It afterward 
meets the horizontal face R"T" along the right line R V, then the cylin- 
drical face g V along a curve projected in V ft, and finally the extrados 
of the arch along the right line ft q-^. All these lines are detejqpined 
by means of horizontal secant planes. By analogous constructions one 
will determine the different lines along which the other joints of the 
arch meet the different parts of the spherical dome. 

For the sake of greater clearness, there are shown on the vertical 
plane the projection A2 Lo N2 B2 of the face curve, of the arch upon 
the spherical dome, as well as of the intradosal edges of the different 
joints, although these projections are not necessary to the preparing 
of the stones. 

60. Cut t ing the Voussoirs i- fake for example the voussoir which 
has for its face in the arch the contour M'N'P'Z' and for its horizontal 



<?.48- 



projection n^^ N Q D E F S. Begin by squaring a right prism (Fig. 117) 
having this projection for a base, and whose height is equal to the 
difference of level of the points ra" and P" (Fig, 116). Then, upon the 
front face of this prism trace the contour Mj Nj Pj^ Zj by means of a 
pattern cut to the contour M'N'P'Z'; mark in the same way upon the 
lateral face the contour Ei Eg J^ % D^ Ci H^ by means of another pat- 
tern cut to the contour h T"R*P"H"M*k. Then, pass through the right 
line El H^ a plane face E^ Hi Ij Ui, perpendicular to the face Ei % Dj 
Jj and such that one can apply to it its appropriate pattern. After- 
ward, cut the cylindrical intrados which passes through Mi Ni, as well 
as the two joints which are contiguous to it, and which pass through the 
right lines Pj N^ and Mr Z, , which can be accomplished by means of a 
square, one branch of which is to be applied upon the face Mi Ni Pi Zi, 
to which that intrados and these joints are perpendicular. When it is 
Judged that these faces have been sufficiently prolonged, apply to them 
their respective patterns. 

The conical Joint Hi Ci Wi Ii will be cut by means of a bevel 
formed by two rectilinear branches containing between them the angle 
h k M" (Fig. 116): while one of the branches glides upon the plane 
Hi Ii Ui El, always resting upon it normal to the curve Hi Ii, the 
other branch will describe the conical Joint, upon which it will be 
necessary afterward to trace the circular arc Ci Wi , which can be done 
by laying off upon this conical face, along several elements, a length 
equal to M"k (Fig. 116). 

The spherical intrados will be cut as directed in treating of the 
spherical dome. , Finally, the upper conical Joint, passing through 
Dl Ki, will be executed by means of a bevel, one branch of which will 
be curved and will form with the other an angle equal to M"N''P" (Fig, 
116), the curved branch having a curvature equal to that of a meridian 
section of the spherical d(»ne. 

In Fig. 118 is represented the voussoir of Fig. 117 seen in its 
natural position. 

61. Descents ;- Descents, like lunettes, may give rise to a Jvuqc- 
tion of voussoirs more or less complex, according as they are right or 
skew with reference to the arch or vault which they penetrate. What 
has been said regarding lunettes should serve to make clear the proper 
treatment of descents. It will often be preferred, however, to avoid 
this constmction and to terminate the descent by a short horizontal 
arch A (Fig. 84) in order to prevent sliding of the voussoirs and the 
bringing of pressure upon the vault penetrated, and the problem will thus 
be reduced to that of the lunette, 

62, Remark ; - Sundry other problems than those treated in the pre- 
ceding pages, and more or less common in architecture, will be found in 
the complete work, Traite Pratique de la Coupe Des Pierres, par snile 

Le J etine , 



-49- 



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Fig-. 9 




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I''i.,. 98. 



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'in. 100. 




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J''ig. 102. 



Fig. 101. 






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Fiy. 108. 




Fig^. 109. 



Kg . 110, 





bx^^\-m\-si»~^>^^-jo^>i».s^"-$^\ J 



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Pi y. 116. 




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